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This program summarises a slice of measurements: how many there are, the smallest and largest, the mean, two kinds of variance and the median. Then it does the same for awkward inputs — repeated values, a single value and no values at all — because a summary that only works on nice data is not finished.",[3411,3440,3441,3442,3445,3446,3422,3450,3422,3453,3432,3456,3435],{},"It is the checkpoint for ",[3419,3443,3444],{"href":889},"Part 18: Algorithms",", and most of the work is done by that part's functions: ",[3447,3448,3449],"code",{},"MinIndex",[3447,3451,3452],{},"MaxIndex",[3447,3454,3455],{},"Sum",[3447,3457,3425],{},[3459,3460,3462],"h2",{"id":3461},"how-it-is-put-together","How it is put together",[3411,3464,3465,3466,3469,3470,3472],{},"One function, ",[3447,3467,3468],{},"Summarize",", does all the work, and ",[3447,3471,1395],{}," calls it four times with different data. Inside, the order of the steps matters:",[3474,3475,3480],"pre",{"className":3476,"code":3477,"language":3478,"meta":3479,"style":3479},"language-mermaid shiki shiki-themes catppuccin-latte catppuccin-latte catppuccin-mocha","flowchart LR\n    s([\"Summarize\"]) --> ext[\"count, smallest, largest\u003Cbr\u002F>(fine for zero values)\"]\n    ext --> empty{\"count == 0?\"}\n    empty -- \"yes\" --> stop([\"stop: nothing to divide\"])\n    empty -- \"no\" --> mean[\"mean\u003Cbr\u002F>first pass\"]\n    mean --> var[\"variance\u003Cbr\u002F>second pass\"]\n    var --> sort[\"Sort, then median\u003Cbr\u002F>(changes the slice)\"]\n","mermaid","",[3447,3481,3482,3490,3496,3502,3508,3514,3520],{"__ignoreMap":3479},[3483,3484,3487],"span",{"class":3485,"line":3486},"line",1,[3483,3488,3489],{},"flowchart LR\n",[3483,3491,3493],{"class":3485,"line":3492},2,[3483,3494,3495],{},"    s([\"Summarize\"]) --> ext[\"count, smallest, largest\u003Cbr\u002F>(fine for zero values)\"]\n",[3483,3497,3499],{"class":3485,"line":3498},3,[3483,3500,3501],{},"    ext --> empty{\"count == 0?\"}\n",[3483,3503,3505],{"class":3485,"line":3504},4,[3483,3506,3507],{},"    empty -- \"yes\" --> stop([\"stop: nothing to divide\"])\n",[3483,3509,3511],{"class":3485,"line":3510},5,[3483,3512,3513],{},"    empty -- \"no\" --> mean[\"mean\u003Cbr\u002F>first pass\"]\n",[3483,3515,3517],{"class":3485,"line":3516},6,[3483,3518,3519],{},"    mean --> var[\"variance\u003Cbr\u002F>second pass\"]\n",[3483,3521,3523],{"class":3485,"line":3522},7,[3483,3524,3525],{},"    var --> sort[\"Sort, then median\u003Cbr\u002F>(changes the slice)\"]\n",[3527,3528,3529,3545],"table",{},[3530,3531,3532],"thead",{},[3533,3534,3535,3539,3542],"tr",{},[3536,3537,3538],"th",{},"Statistic",[3536,3540,3541],{},"How it is found",[3536,3543,3544],{},"Lessons it uses",[3546,3547,3548,3571,3588,3605,3621],"tbody",{},[3533,3549,3550,3554,3564],{},[3551,3552,3553],"td",{},"Smallest, largest",[3551,3555,3556,3422,3558,3560,3561],{},[3447,3557,3449],{},[3447,3559,3452],{}," → ",[3447,3562,3563],{},"uint?",[3551,3565,3566,3422,3568],{},[3419,3567,3428],{"href":908},[3419,3569,3570],{"href":428},"Presence",[3533,3572,3573,3576,3581],{},[3551,3574,3575],{},"Mean",[3551,3577,3578],{},[3447,3579,3580],{},"Sum(values, 0.0) \u002F n",[3551,3582,3583,3422,3585],{},[3419,3584,3431],{"href":912},[3419,3586,3587],{"href":163},"Convert",[3533,3589,3590,3593,3598],{},[3551,3591,3592],{},"Variance, deviation",[3551,3594,3595,3596],{},"A loop of squared distances, ",[3447,3597,2933],{},[3551,3599,3600,3422,3603],{},[3419,3601,3602],{"href":237},"For",[3419,3604,2819],{"href":850},[3533,3606,3607,3610,3615],{},[3551,3608,3609],{},"Median",[3551,3611,3612,3614],{},[3447,3613,3425],{},", then the middle element(s)",[3551,3616,3617,3422,3619],{},[3419,3618,3425],{"href":896},[3419,3620,3421],{"href":311},[3533,3622,3623,3626,3629],{},[3551,3624,3625],{},"Four data sets",[3551,3627,3628],{},"Views of arrays, including an empty one",[3551,3630,3631,3422,3633],{},[3419,3632,3018],{"href":307},[3419,3634,3635],{"href":295},"Array",[3459,3637,3639],{"id":3638},"extremes-that-cope-with-nothing","Extremes that cope with nothing",[3411,3641,3642,3432,3644,3646,3647,3650,3651,3654],{},[3447,3643,3449],{},[3447,3645,3452],{}," do not return a value — they return the ",[3414,3648,3649],{},"position"," of one, as an optional. An empty slice has no smallest element, and ",[3447,3652,3653],{},"none"," says so without any special case:",[3474,3656,3660],{"className":3657,"code":3658,"language":3659,"meta":3479,"style":3479},"language-rux shiki shiki-themes catppuccin-latte catppuccin-latte catppuccin-mocha","match MinIndex(values) {\n    at? => PrintLine(\"  smallest   {}\", values[at]),\n    none => PrintLine(\"  smallest   none\")\n}\n","rux",[3447,3661,3662,3684,3720,3738],{"__ignoreMap":3479},[3483,3663,3664,3667,3671,3675,3678,3681],{"class":3485,"line":3486},[3483,3665,1368],{"class":3666},"smXKV",[3483,3668,3670],{"class":3669},"sJT_S"," MinIndex",[3483,3672,3674],{"class":3673},"sDi1i","(",[3483,3676,3677],{"class":3669},"values",[3483,3679,3680],{"class":3673},")",[3483,3682,3683],{"class":3673}," {\n",[3483,3685,3686,3689,3693,3696,3699,3701,3705,3708,3711,3714,3717],{"class":3485,"line":3492},[3483,3687,3688],{"class":3669},"    at",[3483,3690,3692],{"class":3691},"sCFbw","?",[3483,3694,3695],{"class":3691}," =>",[3483,3697,3698],{"class":3669}," PrintLine",[3483,3700,3674],{"class":3673},[3483,3702,3704],{"class":3703},"sXpww","\"  smallest   {}\"",[3483,3706,3707],{"class":3673},",",[3483,3709,3710],{"class":3669}," values",[3483,3712,3713],{"class":3673},"[",[3483,3715,3716],{"class":3669},"at",[3483,3718,3719],{"class":3673},"]),\n",[3483,3721,3722,3725,3728,3730,3732,3735],{"class":3485,"line":3498},[3483,3723,3724],{"class":3669},"    none ",[3483,3726,3727],{"class":3691},"=>",[3483,3729,3698],{"class":3669},[3483,3731,3674],{"class":3673},[3483,3733,3734],{"class":3703},"\"  smallest   none\"",[3483,3736,3737],{"class":3673},")\n",[3483,3739,3740],{"class":3485,"line":3504},[3483,3741,3742],{"class":3673},"}\n",[3411,3744,3745],{},"Everything after this point divides by the count, so the function stops early when there is nothing to divide:",[3474,3747,3749],{"className":3657,"code":3748,"language":3659,"meta":3479,"style":3479},"if count == 0 {\n    PrintLine(\"  mean, variance and median need at least one value\");\n    PrintLine();\n    return;\n}\n",[3447,3750,3751,3767,3780,3787,3795],{"__ignoreMap":3479},[3483,3752,3753,3755,3758,3761,3765],{"class":3485,"line":3486},[3483,3754,1348],{"class":3666},[3483,3756,3757],{"class":3669}," count ",[3483,3759,3760],{"class":3691},"==",[3483,3762,3764],{"class":3763},"snhBn"," 0",[3483,3766,3683],{"class":3673},[3483,3768,3769,3772,3774,3777],{"class":3485,"line":3492},[3483,3770,3771],{"class":3669},"    PrintLine",[3483,3773,3674],{"class":3673},[3483,3775,3776],{"class":3703},"\"  mean, variance and median need at least one value\"",[3483,3778,3779],{"class":3673},");\n",[3483,3781,3782,3784],{"class":3485,"line":3498},[3483,3783,3771],{"class":3669},[3483,3785,3786],{"class":3673},"();\n",[3483,3788,3789,3792],{"class":3485,"line":3504},[3483,3790,3791],{"class":3666},"    return",[3483,3793,3794],{"class":3673},";\n",[3483,3796,3797],{"class":3485,"line":3510},[3483,3798,3742],{"class":3673},[3459,3800,3802],{"id":3801},"mean-and-variance-two-passes","Mean and variance: two passes",[3411,3804,3805,3806,3809],{},"The ",[3414,3807,3808],{},"variance"," is the average squared distance from the mean, so the mean has to be known before any distance can be measured — hence two passes over the data. The distances are squared so that values above and below the mean both count, instead of cancelling each other out:",[3474,3811,3813],{"className":3657,"code":3812,"language":3659,"meta":3479,"style":3479},"let n = count as float64;\nlet mean = Sum(values, 0.0) \u002F n;\nvar squares = 0.0;\nfor value in values {\n    let distance = value - mean;\n    squares += distance * distance;\n}\n",[3447,3814,3815,3837,3872,3890,3907,3927,3945],{"__ignoreMap":3479},[3483,3816,3817,3820,3823,3826,3828,3831,3835],{"class":3485,"line":3486},[3483,3818,3819],{"class":3666},"let",[3483,3821,3822],{"class":3669}," n ",[3483,3824,3825],{"class":3691},"=",[3483,3827,3757],{"class":3669},[3483,3829,3830],{"class":3691},"as",[3483,3832,3834],{"class":3833},"sqZDx"," float64",[3483,3836,3794],{"class":3673},[3483,3838,3839,3841,3844,3846,3849,3851,3853,3855,3857,3859,3862,3864,3867,3870],{"class":3485,"line":3492},[3483,3840,3819],{"class":3666},[3483,3842,3843],{"class":3669}," mean ",[3483,3845,3825],{"class":3691},[3483,3847,3848],{"class":3669}," Sum",[3483,3850,3674],{"class":3673},[3483,3852,3677],{"class":3669},[3483,3854,3707],{"class":3673},[3483,3856,3764],{"class":3763},[3483,3858,3435],{"class":3673},[3483,3860,3861],{"class":3763},"0",[3483,3863,3680],{"class":3673},[3483,3865,3866],{"class":3691}," \u002F",[3483,3868,3869],{"class":3669}," n",[3483,3871,3794],{"class":3673},[3483,3873,3874,3877,3880,3882,3884,3886,3888],{"class":3485,"line":3498},[3483,3875,3876],{"class":3666},"var",[3483,3878,3879],{"class":3669}," squares ",[3483,3881,3825],{"class":3691},[3483,3883,3764],{"class":3763},[3483,3885,3435],{"class":3673},[3483,3887,3861],{"class":3763},[3483,3889,3794],{"class":3673},[3483,3891,3892,3895,3898,3901,3904],{"class":3485,"line":3504},[3483,3893,3894],{"class":3666},"for",[3483,3896,3897],{"class":3669}," value ",[3483,3899,3900],{"class":3666},"in",[3483,3902,3903],{"class":3669}," values ",[3483,3905,3906],{"class":3673},"{\n",[3483,3908,3909,3912,3915,3917,3919,3922,3925],{"class":3485,"line":3510},[3483,3910,3911],{"class":3666},"    let",[3483,3913,3914],{"class":3669}," distance ",[3483,3916,3825],{"class":3691},[3483,3918,3897],{"class":3669},[3483,3920,3921],{"class":3691},"-",[3483,3923,3924],{"class":3669}," mean",[3483,3926,3794],{"class":3673},[3483,3928,3929,3932,3935,3937,3940,3943],{"class":3485,"line":3516},[3483,3930,3931],{"class":3669},"    squares ",[3483,3933,3934],{"class":3691},"+=",[3483,3936,3914],{"class":3669},[3483,3938,3939],{"class":3691},"*",[3483,3941,3942],{"class":3669}," distance",[3483,3944,3794],{"class":3673},[3483,3946,3947],{"class":3485,"line":3522},[3483,3948,3742],{"class":3673},[3411,3950,3951,3953,3954,3956,3957,3959,3960,3963,3964,3435],{},[3447,3952,3455],{}," is the fold of ",[3447,3955,3117],{}," from the ",[3419,3958,3431],{"href":912}," lesson, already written. Its second argument is the starting value, ",[3447,3961,3962],{},"0.0",", which also fixes the result's type as ",[3447,3965,3966],{},"float64",[3459,3968,3970],{"id":3969},"population-or-sample","Population or sample?",[3411,3972,3973],{},"There are two conventions for \"average squared distance\", and a summary should say which it uses:",[3527,3975,3976,3989],{},[3530,3977,3978],{},[3533,3979,3980,3983,3986],{},[3536,3981,3982],{},"Name",[3536,3984,3985],{},"Divide by",[3536,3987,3988],{},"Use it when",[3546,3990,3991,4002],{},[3533,3992,3993,3996,3999],{},[3551,3994,3995],{},"Population variance",[3551,3997,3998],{},"n",[3551,4000,4001],{},"the data is everything there is",[3533,4003,4004,4007,4010],{},[3551,4005,4006],{},"Sample variance",[3551,4008,4009],{},"n − 1",[3551,4011,4012],{},"the data is a sample of something larger",[3411,4014,4015,4016,4019],{},"Dividing by n − 1 corrects for the sample's mean having been computed from the sample itself. With one value there is nothing to correct with — n − 1 is zero — so the program reports the sample variance as undefined rather than printing a number. The ",[3414,4017,4018],{},"standard deviation"," is the square root of either variance, which brings the answer back into the units of the data.",[3459,4021,4023],{"id":4022},"the-median-sorts-the-data","The median sorts the data",[3411,4025,4026],{},"The median is the middle value once the data is in order, or the average of the two middle values when the count is even:",[3474,4028,4030],{"className":3657,"code":4029,"language":3659,"meta":3479,"style":3479},"Sort(values);\nlet middle = count \u002F 2;\nif count % 2 == 1 {\n    PrintLine(\"  median     {} (the middle value)\", values[middle]);\n} else {\n    let low = values[middle - 1];\n    let high = values[middle];\n    PrintLine(\"  median     {} (between {} and {})\", (low + high) \u002F 2.0, low, high);\n}\n",[3447,4031,4032,4042,4061,4080,4101,4111,4134,4151,4195],{"__ignoreMap":3479},[3483,4033,4034,4036,4038,4040],{"class":3485,"line":3486},[3483,4035,3425],{"class":3669},[3483,4037,3674],{"class":3673},[3483,4039,3677],{"class":3669},[3483,4041,3779],{"class":3673},[3483,4043,4044,4046,4049,4051,4053,4056,4059],{"class":3485,"line":3492},[3483,4045,3819],{"class":3666},[3483,4047,4048],{"class":3669}," middle ",[3483,4050,3825],{"class":3691},[3483,4052,3757],{"class":3669},[3483,4054,4055],{"class":3691},"\u002F",[3483,4057,4058],{"class":3763}," 2",[3483,4060,3794],{"class":3673},[3483,4062,4063,4065,4067,4070,4072,4075,4078],{"class":3485,"line":3498},[3483,4064,1348],{"class":3666},[3483,4066,3757],{"class":3669},[3483,4068,4069],{"class":3691},"%",[3483,4071,4058],{"class":3763},[3483,4073,4074],{"class":3691}," ==",[3483,4076,4077],{"class":3763}," 1",[3483,4079,3683],{"class":3673},[3483,4081,4082,4084,4086,4089,4091,4093,4095,4098],{"class":3485,"line":3504},[3483,4083,3771],{"class":3669},[3483,4085,3674],{"class":3673},[3483,4087,4088],{"class":3703},"\"  median     {} (the middle value)\"",[3483,4090,3707],{"class":3673},[3483,4092,3710],{"class":3669},[3483,4094,3713],{"class":3673},[3483,4096,4097],{"class":3669},"middle",[3483,4099,4100],{"class":3673},"]);\n",[3483,4102,4103,4106,4109],{"class":3485,"line":3510},[3483,4104,4105],{"class":3673},"}",[3483,4107,4108],{"class":3666}," else",[3483,4110,3683],{"class":3673},[3483,4112,4113,4115,4118,4120,4122,4124,4127,4129,4131],{"class":3485,"line":3516},[3483,4114,3911],{"class":3666},[3483,4116,4117],{"class":3669}," low ",[3483,4119,3825],{"class":3691},[3483,4121,3710],{"class":3669},[3483,4123,3713],{"class":3673},[3483,4125,4126],{"class":3669},"middle ",[3483,4128,3921],{"class":3691},[3483,4130,4077],{"class":3763},[3483,4132,4133],{"class":3673},"];\n",[3483,4135,4136,4138,4141,4143,4145,4147,4149],{"class":3485,"line":3522},[3483,4137,3911],{"class":3666},[3483,4139,4140],{"class":3669}," high ",[3483,4142,3825],{"class":3691},[3483,4144,3710],{"class":3669},[3483,4146,3713],{"class":3673},[3483,4148,4097],{"class":3669},[3483,4150,4133],{"class":3673},[3483,4152,4154,4156,4158,4161,4163,4166,4169,4171,4174,4176,4178,4180,4182,4184,4186,4189,4191,4193],{"class":3485,"line":4153},8,[3483,4155,3771],{"class":3669},[3483,4157,3674],{"class":3673},[3483,4159,4160],{"class":3703},"\"  median     {} (between {} and {})\"",[3483,4162,3707],{"class":3673},[3483,4164,4165],{"class":3673}," (",[3483,4167,4168],{"class":3669},"low ",[3483,4170,3117],{"class":3691},[3483,4172,4173],{"class":3669}," high",[3483,4175,3680],{"class":3673},[3483,4177,3866],{"class":3691},[3483,4179,4058],{"class":3763},[3483,4181,3435],{"class":3673},[3483,4183,3861],{"class":3763},[3483,4185,3707],{"class":3673},[3483,4187,4188],{"class":3669}," low",[3483,4190,3707],{"class":3673},[3483,4192,4173],{"class":3669},[3483,4194,3779],{"class":3673},[3483,4196,4198],{"class":3485,"line":4197},9,[3483,4199,3742],{"class":3673},[3411,4201,4202,4204,4205,4207,4208,4211,4212,4215,4216,4218],{},[3447,4203,3425],{}," works in place, so ",[3447,4206,3468],{}," takes a ",[3414,4209,4210],{},"writable"," slice, ",[3447,4213,4214],{},"var float64[..]",", and the caller's array is left sorted afterwards. That is why the median comes last: the ",[3447,4217,3677],{}," line at the top of each summary still shows the data in its original order.",[3459,4220,4222],{"id":4221},"four-kinds-of-input","Four kinds of input",[3411,4224,4225,4227,4228,4230,4231,4234],{},[3447,4226,1395],{}," keeps each data set in a ",[3447,4229,3876],{}," array and passes a view of it. The last call passes ",[3447,4232,4233],{},"readings[..0]",", an empty view of an array that already exists — the simplest way to make \"no values\" without a new type:",[3474,4236,4238],{"className":3657,"code":4237,"language":3659,"meta":3479,"style":3479},"var single: float64[1] = [42.0];\nSummarize(\"a single value\", single[..]);\n\n\u002F\u002F An empty view of an array, so there is nothing to summarise.\nSummarize(\"no values\", readings[..0]);\n",[3447,4239,4240,4275,4295,4301,4307],{"__ignoreMap":3479},[3483,4241,4242,4244,4247,4250,4252,4254,4257,4260,4263,4266,4269,4271,4273],{"class":3485,"line":3486},[3483,4243,3876],{"class":3666},[3483,4245,4246],{"class":3669}," single",[3483,4248,4249],{"class":3673},":",[3483,4251,3834],{"class":3833},[3483,4253,3713],{"class":3673},[3483,4255,4256],{"class":3763},"1",[3483,4258,4259],{"class":3673},"]",[3483,4261,4262],{"class":3691}," =",[3483,4264,4265],{"class":3673}," [",[3483,4267,4268],{"class":3763},"42",[3483,4270,3435],{"class":3673},[3483,4272,3861],{"class":3763},[3483,4274,4133],{"class":3673},[3483,4276,4277,4279,4281,4284,4286,4288,4290,4293],{"class":3485,"line":3492},[3483,4278,3468],{"class":3669},[3483,4280,3674],{"class":3673},[3483,4282,4283],{"class":3703},"\"a single value\"",[3483,4285,3707],{"class":3673},[3483,4287,4246],{"class":3669},[3483,4289,3713],{"class":3673},[3483,4291,4292],{"class":3691},"..",[3483,4294,4100],{"class":3673},[3483,4296,4297],{"class":3485,"line":3498},[3483,4298,4300],{"emptyLinePlaceholder":4299},true,"\n",[3483,4302,4303],{"class":3485,"line":3504},[3483,4304,4306],{"class":4305},"sY9YT","\u002F\u002F An empty view of an array, so there is nothing to summarise.\n",[3483,4308,4309,4311,4313,4316,4318,4321,4323,4325,4327],{"class":3485,"line":3510},[3483,4310,3468],{"class":3669},[3483,4312,3674],{"class":3673},[3483,4314,4315],{"class":3703},"\"no values\"",[3483,4317,3707],{"class":3673},[3483,4319,4320],{"class":3669}," readings",[3483,4322,3713],{"class":3673},[3483,4324,4292],{"class":3691},[3483,4326,3861],{"class":3763},[3483,4328,4100],{"class":3673},[3459,4330,4332],{"id":4331},"the-program","The program",[3411,4334,4335,4336,4342],{},"The whole lesson is one package in the ",[3419,4337,4341],{"href":4338,"rel":4339},"https:\u002F\u002Fgithub.com\u002Frux-lang\u002FExamples\u002Ftree\u002Fmain\u002FProjects\u002FStatistics",[4340],"nofollow","Examples repository",". Its comments explain every step.",[3474,4344,4347],{"className":3657,"code":4345,"filename":4346,"language":3659,"meta":3479,"style":3479},"\u002F\u002F Summarising a set of numbers: where the middle is, and how spread out the numbers are around it.\n\u002F\u002F\n\u002F\u002F The mean is the total divided by the count. The variance is the average squared distance from\n\u002F\u002F the mean, and the standard deviation is its square root, which brings the answer back to the\n\u002F\u002F units of the data. There are two conventions for that average, and a summary should say which\n\u002F\u002F one it uses:\n\u002F\u002F\n\u002F\u002F     population variance   divide by n       the data is everything there is\n\u002F\u002F     sample variance       divide by n - 1   the data is a sample of something larger\n\u002F\u002F\n\u002F\u002F Dividing by n - 1 corrects for the sample's mean being computed from the sample itself. With one\n\u002F\u002F value there is nothing to correct with, so the sample variance is undefined rather than zero.\n\u002F\u002F\n\u002F\u002F The median is the middle value once the data is sorted, or the average of the two middle values\n\u002F\u002F when the count is even. `Sort` works in place, so `Summarize` takes a writable slice and leaves\n\u002F\u002F the caller's data in order.\nimport Algorithms::{ MaxIndex, MinIndex, Sort, Sum };\nimport Io::{ Print, PrintLine };\nimport Math::Sqrt;\n\nfunc Summarize(label: char8[..], values: var float64[..]) {\n    PrintLine(\"{}\", label);\n    Print(\"  values    \");\n    for value in values {\n        Print(\" {}\", value);\n    }\n    PrintLine();\n    let count = values.length;\n    PrintLine(\"  count      {}\", count);\n\n    \u002F\u002F The extremes answer with an optional index, so an empty slice is handled right here.\n    match MinIndex(values) {\n        at? => PrintLine(\"  smallest   {}\", values[at]),\n        none => PrintLine(\"  smallest   none\")\n    }\n    match MaxIndex(values) {\n        at? => PrintLine(\"  largest    {}\", values[at]),\n        none => PrintLine(\"  largest    none\")\n    }\n\n    \u002F\u002F Everything else divides by the count, and there is no mean of nothing.\n    if count == 0 {\n        PrintLine(\"  mean, variance and median need at least one value\");\n        PrintLine();\n        return;\n    }\n\n    \u002F\u002F Two passes: the mean has to be known before the distances from it can be measured. The\n    \u002F\u002F distances are squared so that values above and below both count, instead of cancelling.\n    \u002F\u002F `Sum` is the fold of `+` from the Fold lesson, already written.\n    let n = count as float64;\n    let mean = Sum(values, 0.0) \u002F n;\n    var squares = 0.0;\n    for value in values {\n        let distance = value - mean;\n        squares += distance * distance;\n    }\n    PrintLine(\"  mean       {:.4}\", mean);\n\n    let population = squares \u002F n;\n    PrintLine(\"  population variance {:.4}, deviation {:.4}\", population, Sqrt(population));\n    if count > 1 {\n        let sample = squares \u002F (n - 1.0);\n        PrintLine(\"  sample     variance {:.4}, deviation {:.4}\", sample, Sqrt(sample));\n    } else {\n        PrintLine(\"  sample     variance undefined for one value\");\n    }\n\n    \u002F\u002F The median needs the values in order, so it comes last.\n    Sort(values);\n    let middle = count \u002F 2;\n    if count % 2 == 1 {\n        PrintLine(\"  median     {} (the middle value)\", values[middle]);\n    } else {\n        let low = values[middle - 1];\n        let high = values[middle];\n        PrintLine(\"  median     {} (between {} and {})\", (low + high) \u002F 2.0, low, high);\n    }\n    PrintLine();\n}\n\nfunc Main() -> int {\n    \u002F\u002F An odd count: one value sits exactly in the middle.\n    var readings: float64[9] = [12.5, 9.0, 15.25, 11.0, 8.75, 14.0, 10.5, 13.25, 11.75];\n    Summarize(\"nine readings\", readings[..]);\n\n    \u002F\u002F An even count with repeated values. Duplicates are ordinary data: each one counts.\n    var scores: float64[6] = [4.0, 7.0, 4.0, 1.0, 7.0, 7.0];\n    Summarize(\"six scores with repeats\", scores[..]);\n\n    \u002F\u002F One value: no spread at all, and no sample variance.\n    var single: float64[1] = [42.0];\n    Summarize(\"a single value\", single[..]);\n\n    \u002F\u002F An empty view of an array, so there is nothing to summarise.\n    Summarize(\"no values\", readings[..0]);\n    return 0;\n}\n","Src\u002FMain.rux",[3447,4348,4349,4354,4359,4364,4369,4374,4379,4383,4388,4393,4398,4404,4410,4415,4421,4427,4433,4468,4490,4504,4509,4550,4567,4580,4594,4612,4618,4625,4643,4660,4665,4671,4687,4713,4729,4734,4749,4775,4791,4796,4801,4807,4821,4833,4840,4848,4853,4858,4864,4870,4876,4893,4924,4942,4955,4973,4989,4994,5010,5015,5033,5061,5075,5104,5130,5140,5152,5157,5162,5168,5180,5197,5214,5233,5242,5263,5280,5319,5324,5331,5336,5341,5360,5366,5471,5492,5497,5503,5577,5597,5602,5608,5637,5656,5661,5667,5688,5697],{"__ignoreMap":3479},[3483,4350,4351],{"class":3485,"line":3486},[3483,4352,4353],{"class":4305},"\u002F\u002F Summarising a set of numbers: where the middle is, and how spread out the numbers are around it.\n",[3483,4355,4356],{"class":3485,"line":3492},[3483,4357,4358],{"class":4305},"\u002F\u002F\n",[3483,4360,4361],{"class":3485,"line":3498},[3483,4362,4363],{"class":4305},"\u002F\u002F The mean is the total divided by the count. The variance is the average squared distance from\n",[3483,4365,4366],{"class":3485,"line":3504},[3483,4367,4368],{"class":4305},"\u002F\u002F the mean, and the standard deviation is its square root, which brings the answer back to the\n",[3483,4370,4371],{"class":3485,"line":3510},[3483,4372,4373],{"class":4305},"\u002F\u002F units of the data. There are two conventions for that average, and a summary should say which\n",[3483,4375,4376],{"class":3485,"line":3516},[3483,4377,4378],{"class":4305},"\u002F\u002F one it uses:\n",[3483,4380,4381],{"class":3485,"line":3522},[3483,4382,4358],{"class":4305},[3483,4384,4385],{"class":3485,"line":4153},[3483,4386,4387],{"class":4305},"\u002F\u002F     population variance   divide by n       the data is everything there is\n",[3483,4389,4390],{"class":3485,"line":4197},[3483,4391,4392],{"class":4305},"\u002F\u002F     sample variance       divide by n - 1   the data is a sample of something larger\n",[3483,4394,4396],{"class":3485,"line":4395},10,[3483,4397,4358],{"class":4305},[3483,4399,4401],{"class":3485,"line":4400},11,[3483,4402,4403],{"class":4305},"\u002F\u002F Dividing by n - 1 corrects for the sample's mean being computed from the sample itself. With one\n",[3483,4405,4407],{"class":3485,"line":4406},12,[3483,4408,4409],{"class":4305},"\u002F\u002F value there is nothing to correct with, so the sample variance is undefined rather than zero.\n",[3483,4411,4413],{"class":3485,"line":4412},13,[3483,4414,4358],{"class":4305},[3483,4416,4418],{"class":3485,"line":4417},14,[3483,4419,4420],{"class":4305},"\u002F\u002F The median is the middle value once the data is sorted, or the average of the two middle values\n",[3483,4422,4424],{"class":3485,"line":4423},15,[3483,4425,4426],{"class":4305},"\u002F\u002F when the count is even. `Sort` works in place, so `Summarize` takes a writable slice and leaves\n",[3483,4428,4430],{"class":3485,"line":4429},16,[3483,4431,4432],{"class":4305},"\u002F\u002F the caller's data in order.\n",[3483,4434,4436,4439,4442,4445,4448,4451,4453,4455,4457,4460,4462,4465],{"class":3485,"line":4435},17,[3483,4437,4438],{"class":3666},"import",[3483,4440,4441],{"class":3669}," Algorithms",[3483,4443,4444],{"class":3691},"::",[3483,4446,4447],{"class":3673},"{",[3483,4449,4450],{"class":3669}," MaxIndex",[3483,4452,3707],{"class":3673},[3483,4454,3670],{"class":3669},[3483,4456,3707],{"class":3673},[3483,4458,4459],{"class":3669}," Sort",[3483,4461,3707],{"class":3673},[3483,4463,4464],{"class":3669}," Sum ",[3483,4466,4467],{"class":3673},"};\n",[3483,4469,4471,4473,4476,4478,4480,4483,4485,4488],{"class":3485,"line":4470},18,[3483,4472,4438],{"class":3666},[3483,4474,4475],{"class":3669}," Io",[3483,4477,4444],{"class":3691},[3483,4479,4447],{"class":3673},[3483,4481,4482],{"class":3669}," Print",[3483,4484,3707],{"class":3673},[3483,4486,4487],{"class":3669}," PrintLine ",[3483,4489,4467],{"class":3673},[3483,4491,4493,4495,4498,4500,4502],{"class":3485,"line":4492},19,[3483,4494,4438],{"class":3666},[3483,4496,4497],{"class":3669}," Math",[3483,4499,4444],{"class":3691},[3483,4501,2933],{"class":3669},[3483,4503,3794],{"class":3673},[3483,4505,4507],{"class":3485,"line":4506},20,[3483,4508,4300],{"emptyLinePlaceholder":4299},[3483,4510,4512,4515,4519,4521,4525,4528,4531,4534,4536,4538,4540,4542,4544,4546,4548],{"class":3485,"line":4511},21,[3483,4513,4514],{"class":3666},"func",[3483,4516,4518],{"class":4517},"sptzg"," Summarize",[3483,4520,3674],{"class":3673},[3483,4522,4524],{"class":4523},"sYR1E","label",[3483,4526,4527],{"class":3673},": ",[3483,4529,4530],{"class":3833},"char8",[3483,4532,4533],{"class":3669},"[..]",[3483,4535,3707],{"class":3673},[3483,4537,3710],{"class":4523},[3483,4539,4527],{"class":3673},[3483,4541,3876],{"class":3833},[3483,4543,3834],{"class":3833},[3483,4545,4533],{"class":3669},[3483,4547,3680],{"class":3673},[3483,4549,3683],{"class":3673},[3483,4551,4553,4555,4557,4560,4562,4565],{"class":3485,"line":4552},22,[3483,4554,3771],{"class":3669},[3483,4556,3674],{"class":3673},[3483,4558,4559],{"class":3703},"\"{}\"",[3483,4561,3707],{"class":3673},[3483,4563,4564],{"class":3669}," label",[3483,4566,3779],{"class":3673},[3483,4568,4570,4573,4575,4578],{"class":3485,"line":4569},23,[3483,4571,4572],{"class":3669},"    Print",[3483,4574,3674],{"class":3673},[3483,4576,4577],{"class":3703},"\"  values    \"",[3483,4579,3779],{"class":3673},[3483,4581,4583,4586,4588,4590,4592],{"class":3485,"line":4582},24,[3483,4584,4585],{"class":3666},"    for",[3483,4587,3897],{"class":3669},[3483,4589,3900],{"class":3666},[3483,4591,3903],{"class":3669},[3483,4593,3906],{"class":3673},[3483,4595,4597,4600,4602,4605,4607,4610],{"class":3485,"line":4596},25,[3483,4598,4599],{"class":3669},"        Print",[3483,4601,3674],{"class":3673},[3483,4603,4604],{"class":3703},"\" {}\"",[3483,4606,3707],{"class":3673},[3483,4608,4609],{"class":3669}," value",[3483,4611,3779],{"class":3673},[3483,4613,4615],{"class":3485,"line":4614},26,[3483,4616,4617],{"class":3673},"    }\n",[3483,4619,4621,4623],{"class":3485,"line":4620},27,[3483,4622,3771],{"class":3669},[3483,4624,3786],{"class":3673},[3483,4626,4628,4630,4632,4634,4636,4638,4641],{"class":3485,"line":4627},28,[3483,4629,3911],{"class":3666},[3483,4631,3757],{"class":3669},[3483,4633,3825],{"class":3691},[3483,4635,3710],{"class":3669},[3483,4637,3435],{"class":3691},[3483,4639,4640],{"class":3669},"length",[3483,4642,3794],{"class":3673},[3483,4644,4646,4648,4650,4653,4655,4658],{"class":3485,"line":4645},29,[3483,4647,3771],{"class":3669},[3483,4649,3674],{"class":3673},[3483,4651,4652],{"class":3703},"\"  count      {}\"",[3483,4654,3707],{"class":3673},[3483,4656,4657],{"class":3669}," count",[3483,4659,3779],{"class":3673},[3483,4661,4663],{"class":3485,"line":4662},30,[3483,4664,4300],{"emptyLinePlaceholder":4299},[3483,4666,4668],{"class":3485,"line":4667},31,[3483,4669,4670],{"class":4305},"    \u002F\u002F The extremes answer with an optional index, so an empty slice is handled right here.\n",[3483,4672,4674,4677,4679,4681,4683,4685],{"class":3485,"line":4673},32,[3483,4675,4676],{"class":3666},"    match",[3483,4678,3670],{"class":3669},[3483,4680,3674],{"class":3673},[3483,4682,3677],{"class":3669},[3483,4684,3680],{"class":3673},[3483,4686,3683],{"class":3673},[3483,4688,4690,4693,4695,4697,4699,4701,4703,4705,4707,4709,4711],{"class":3485,"line":4689},33,[3483,4691,4692],{"class":3669},"        at",[3483,4694,3692],{"class":3691},[3483,4696,3695],{"class":3691},[3483,4698,3698],{"class":3669},[3483,4700,3674],{"class":3673},[3483,4702,3704],{"class":3703},[3483,4704,3707],{"class":3673},[3483,4706,3710],{"class":3669},[3483,4708,3713],{"class":3673},[3483,4710,3716],{"class":3669},[3483,4712,3719],{"class":3673},[3483,4714,4716,4719,4721,4723,4725,4727],{"class":3485,"line":4715},34,[3483,4717,4718],{"class":3669},"        none ",[3483,4720,3727],{"class":3691},[3483,4722,3698],{"class":3669},[3483,4724,3674],{"class":3673},[3483,4726,3734],{"class":3703},[3483,4728,3737],{"class":3673},[3483,4730,4732],{"class":3485,"line":4731},35,[3483,4733,4617],{"class":3673},[3483,4735,4737,4739,4741,4743,4745,4747],{"class":3485,"line":4736},36,[3483,4738,4676],{"class":3666},[3483,4740,4450],{"class":3669},[3483,4742,3674],{"class":3673},[3483,4744,3677],{"class":3669},[3483,4746,3680],{"class":3673},[3483,4748,3683],{"class":3673},[3483,4750,4752,4754,4756,4758,4760,4762,4765,4767,4769,4771,4773],{"class":3485,"line":4751},37,[3483,4753,4692],{"class":3669},[3483,4755,3692],{"class":3691},[3483,4757,3695],{"class":3691},[3483,4759,3698],{"class":3669},[3483,4761,3674],{"class":3673},[3483,4763,4764],{"class":3703},"\"  largest    {}\"",[3483,4766,3707],{"class":3673},[3483,4768,3710],{"class":3669},[3483,4770,3713],{"class":3673},[3483,4772,3716],{"class":3669},[3483,4774,3719],{"class":3673},[3483,4776,4778,4780,4782,4784,4786,4789],{"class":3485,"line":4777},38,[3483,4779,4718],{"class":3669},[3483,4781,3727],{"class":3691},[3483,4783,3698],{"class":3669},[3483,4785,3674],{"class":3673},[3483,4787,4788],{"class":3703},"\"  largest    none\"",[3483,4790,3737],{"class":3673},[3483,4792,4794],{"class":3485,"line":4793},39,[3483,4795,4617],{"class":3673},[3483,4797,4799],{"class":3485,"line":4798},40,[3483,4800,4300],{"emptyLinePlaceholder":4299},[3483,4802,4804],{"class":3485,"line":4803},41,[3483,4805,4806],{"class":4305},"    \u002F\u002F Everything else divides by the count, and there is no mean of nothing.\n",[3483,4808,4810,4813,4815,4817,4819],{"class":3485,"line":4809},42,[3483,4811,4812],{"class":3666},"    if",[3483,4814,3757],{"class":3669},[3483,4816,3760],{"class":3691},[3483,4818,3764],{"class":3763},[3483,4820,3683],{"class":3673},[3483,4822,4824,4827,4829,4831],{"class":3485,"line":4823},43,[3483,4825,4826],{"class":3669},"        PrintLine",[3483,4828,3674],{"class":3673},[3483,4830,3776],{"class":3703},[3483,4832,3779],{"class":3673},[3483,4834,4836,4838],{"class":3485,"line":4835},44,[3483,4837,4826],{"class":3669},[3483,4839,3786],{"class":3673},[3483,4841,4843,4846],{"class":3485,"line":4842},45,[3483,4844,4845],{"class":3666},"        return",[3483,4847,3794],{"class":3673},[3483,4849,4851],{"class":3485,"line":4850},46,[3483,4852,4617],{"class":3673},[3483,4854,4856],{"class":3485,"line":4855},47,[3483,4857,4300],{"emptyLinePlaceholder":4299},[3483,4859,4861],{"class":3485,"line":4860},48,[3483,4862,4863],{"class":4305},"    \u002F\u002F Two passes: the mean has to be known before the distances from it can be measured. The\n",[3483,4865,4867],{"class":3485,"line":4866},49,[3483,4868,4869],{"class":4305},"    \u002F\u002F distances are squared so that values above and below both count, instead of cancelling.\n",[3483,4871,4873],{"class":3485,"line":4872},50,[3483,4874,4875],{"class":4305},"    \u002F\u002F `Sum` is the fold of `+` from the Fold lesson, already written.\n",[3483,4877,4879,4881,4883,4885,4887,4889,4891],{"class":3485,"line":4878},51,[3483,4880,3911],{"class":3666},[3483,4882,3822],{"class":3669},[3483,4884,3825],{"class":3691},[3483,4886,3757],{"class":3669},[3483,4888,3830],{"class":3691},[3483,4890,3834],{"class":3833},[3483,4892,3794],{"class":3673},[3483,4894,4896,4898,4900,4902,4904,4906,4908,4910,4912,4914,4916,4918,4920,4922],{"class":3485,"line":4895},52,[3483,4897,3911],{"class":3666},[3483,4899,3843],{"class":3669},[3483,4901,3825],{"class":3691},[3483,4903,3848],{"class":3669},[3483,4905,3674],{"class":3673},[3483,4907,3677],{"class":3669},[3483,4909,3707],{"class":3673},[3483,4911,3764],{"class":3763},[3483,4913,3435],{"class":3673},[3483,4915,3861],{"class":3763},[3483,4917,3680],{"class":3673},[3483,4919,3866],{"class":3691},[3483,4921,3869],{"class":3669},[3483,4923,3794],{"class":3673},[3483,4925,4927,4930,4932,4934,4936,4938,4940],{"class":3485,"line":4926},53,[3483,4928,4929],{"class":3666},"    var",[3483,4931,3879],{"class":3669},[3483,4933,3825],{"class":3691},[3483,4935,3764],{"class":3763},[3483,4937,3435],{"class":3673},[3483,4939,3861],{"class":3763},[3483,4941,3794],{"class":3673},[3483,4943,4945,4947,4949,4951,4953],{"class":3485,"line":4944},54,[3483,4946,4585],{"class":3666},[3483,4948,3897],{"class":3669},[3483,4950,3900],{"class":3666},[3483,4952,3903],{"class":3669},[3483,4954,3906],{"class":3673},[3483,4956,4958,4961,4963,4965,4967,4969,4971],{"class":3485,"line":4957},55,[3483,4959,4960],{"class":3666},"        let",[3483,4962,3914],{"class":3669},[3483,4964,3825],{"class":3691},[3483,4966,3897],{"class":3669},[3483,4968,3921],{"class":3691},[3483,4970,3924],{"class":3669},[3483,4972,3794],{"class":3673},[3483,4974,4976,4979,4981,4983,4985,4987],{"class":3485,"line":4975},56,[3483,4977,4978],{"class":3669},"        squares ",[3483,4980,3934],{"class":3691},[3483,4982,3914],{"class":3669},[3483,4984,3939],{"class":3691},[3483,4986,3942],{"class":3669},[3483,4988,3794],{"class":3673},[3483,4990,4992],{"class":3485,"line":4991},57,[3483,4993,4617],{"class":3673},[3483,4995,4997,4999,5001,5004,5006,5008],{"class":3485,"line":4996},58,[3483,4998,3771],{"class":3669},[3483,5000,3674],{"class":3673},[3483,5002,5003],{"class":3703},"\"  mean       {:.4}\"",[3483,5005,3707],{"class":3673},[3483,5007,3924],{"class":3669},[3483,5009,3779],{"class":3673},[3483,5011,5013],{"class":3485,"line":5012},59,[3483,5014,4300],{"emptyLinePlaceholder":4299},[3483,5016,5018,5020,5023,5025,5027,5029,5031],{"class":3485,"line":5017},60,[3483,5019,3911],{"class":3666},[3483,5021,5022],{"class":3669}," population ",[3483,5024,3825],{"class":3691},[3483,5026,3879],{"class":3669},[3483,5028,4055],{"class":3691},[3483,5030,3869],{"class":3669},[3483,5032,3794],{"class":3673},[3483,5034,5036,5038,5040,5043,5045,5048,5050,5053,5055,5058],{"class":3485,"line":5035},61,[3483,5037,3771],{"class":3669},[3483,5039,3674],{"class":3673},[3483,5041,5042],{"class":3703},"\"  population variance {:.4}, deviation {:.4}\"",[3483,5044,3707],{"class":3673},[3483,5046,5047],{"class":3669}," population",[3483,5049,3707],{"class":3673},[3483,5051,5052],{"class":3669}," Sqrt",[3483,5054,3674],{"class":3673},[3483,5056,5057],{"class":3669},"population",[3483,5059,5060],{"class":3673},"));\n",[3483,5062,5064,5066,5068,5071,5073],{"class":3485,"line":5063},62,[3483,5065,4812],{"class":3666},[3483,5067,3757],{"class":3669},[3483,5069,5070],{"class":3691},">",[3483,5072,4077],{"class":3763},[3483,5074,3683],{"class":3673},[3483,5076,5078,5080,5083,5085,5087,5089,5091,5094,5096,5098,5100,5102],{"class":3485,"line":5077},63,[3483,5079,4960],{"class":3666},[3483,5081,5082],{"class":3669}," sample ",[3483,5084,3825],{"class":3691},[3483,5086,3879],{"class":3669},[3483,5088,4055],{"class":3691},[3483,5090,4165],{"class":3673},[3483,5092,5093],{"class":3669},"n ",[3483,5095,3921],{"class":3691},[3483,5097,4077],{"class":3763},[3483,5099,3435],{"class":3673},[3483,5101,3861],{"class":3763},[3483,5103,3779],{"class":3673},[3483,5105,5107,5109,5111,5114,5116,5119,5121,5123,5125,5128],{"class":3485,"line":5106},64,[3483,5108,4826],{"class":3669},[3483,5110,3674],{"class":3673},[3483,5112,5113],{"class":3703},"\"  sample     variance {:.4}, deviation {:.4}\"",[3483,5115,3707],{"class":3673},[3483,5117,5118],{"class":3669}," sample",[3483,5120,3707],{"class":3673},[3483,5122,5052],{"class":3669},[3483,5124,3674],{"class":3673},[3483,5126,5127],{"class":3669},"sample",[3483,5129,5060],{"class":3673},[3483,5131,5133,5136,5138],{"class":3485,"line":5132},65,[3483,5134,5135],{"class":3673},"    }",[3483,5137,4108],{"class":3666},[3483,5139,3683],{"class":3673},[3483,5141,5143,5145,5147,5150],{"class":3485,"line":5142},66,[3483,5144,4826],{"class":3669},[3483,5146,3674],{"class":3673},[3483,5148,5149],{"class":3703},"\"  sample     variance undefined for one value\"",[3483,5151,3779],{"class":3673},[3483,5153,5155],{"class":3485,"line":5154},67,[3483,5156,4617],{"class":3673},[3483,5158,5160],{"class":3485,"line":5159},68,[3483,5161,4300],{"emptyLinePlaceholder":4299},[3483,5163,5165],{"class":3485,"line":5164},69,[3483,5166,5167],{"class":4305},"    \u002F\u002F The median needs the values in order, so it comes last.\n",[3483,5169,5171,5174,5176,5178],{"class":3485,"line":5170},70,[3483,5172,5173],{"class":3669},"    Sort",[3483,5175,3674],{"class":3673},[3483,5177,3677],{"class":3669},[3483,5179,3779],{"class":3673},[3483,5181,5183,5185,5187,5189,5191,5193,5195],{"class":3485,"line":5182},71,[3483,5184,3911],{"class":3666},[3483,5186,4048],{"class":3669},[3483,5188,3825],{"class":3691},[3483,5190,3757],{"class":3669},[3483,5192,4055],{"class":3691},[3483,5194,4058],{"class":3763},[3483,5196,3794],{"class":3673},[3483,5198,5200,5202,5204,5206,5208,5210,5212],{"class":3485,"line":5199},72,[3483,5201,4812],{"class":3666},[3483,5203,3757],{"class":3669},[3483,5205,4069],{"class":3691},[3483,5207,4058],{"class":3763},[3483,5209,4074],{"class":3691},[3483,5211,4077],{"class":3763},[3483,5213,3683],{"class":3673},[3483,5215,5217,5219,5221,5223,5225,5227,5229,5231],{"class":3485,"line":5216},73,[3483,5218,4826],{"class":3669},[3483,5220,3674],{"class":3673},[3483,5222,4088],{"class":3703},[3483,5224,3707],{"class":3673},[3483,5226,3710],{"class":3669},[3483,5228,3713],{"class":3673},[3483,5230,4097],{"class":3669},[3483,5232,4100],{"class":3673},[3483,5234,5236,5238,5240],{"class":3485,"line":5235},74,[3483,5237,5135],{"class":3673},[3483,5239,4108],{"class":3666},[3483,5241,3683],{"class":3673},[3483,5243,5245,5247,5249,5251,5253,5255,5257,5259,5261],{"class":3485,"line":5244},75,[3483,5246,4960],{"class":3666},[3483,5248,4117],{"class":3669},[3483,5250,3825],{"class":3691},[3483,5252,3710],{"class":3669},[3483,5254,3713],{"class":3673},[3483,5256,4126],{"class":3669},[3483,5258,3921],{"class":3691},[3483,5260,4077],{"class":3763},[3483,5262,4133],{"class":3673},[3483,5264,5266,5268,5270,5272,5274,5276,5278],{"class":3485,"line":5265},76,[3483,5267,4960],{"class":3666},[3483,5269,4140],{"class":3669},[3483,5271,3825],{"class":3691},[3483,5273,3710],{"class":3669},[3483,5275,3713],{"class":3673},[3483,5277,4097],{"class":3669},[3483,5279,4133],{"class":3673},[3483,5281,5283,5285,5287,5289,5291,5293,5295,5297,5299,5301,5303,5305,5307,5309,5311,5313,5315,5317],{"class":3485,"line":5282},77,[3483,5284,4826],{"class":3669},[3483,5286,3674],{"class":3673},[3483,5288,4160],{"class":3703},[3483,5290,3707],{"class":3673},[3483,5292,4165],{"class":3673},[3483,5294,4168],{"class":3669},[3483,5296,3117],{"class":3691},[3483,5298,4173],{"class":3669},[3483,5300,3680],{"class":3673},[3483,5302,3866],{"class":3691},[3483,5304,4058],{"class":3763},[3483,5306,3435],{"class":3673},[3483,5308,3861],{"class":3763},[3483,5310,3707],{"class":3673},[3483,5312,4188],{"class":3669},[3483,5314,3707],{"class":3673},[3483,5316,4173],{"class":3669},[3483,5318,3779],{"class":3673},[3483,5320,5322],{"class":3485,"line":5321},78,[3483,5323,4617],{"class":3673},[3483,5325,5327,5329],{"class":3485,"line":5326},79,[3483,5328,3771],{"class":3669},[3483,5330,3786],{"class":3673},[3483,5332,5334],{"class":3485,"line":5333},80,[3483,5335,3742],{"class":3673},[3483,5337,5339],{"class":3485,"line":5338},81,[3483,5340,4300],{"emptyLinePlaceholder":4299},[3483,5342,5344,5346,5349,5352,5355,5358],{"class":3485,"line":5343},82,[3483,5345,4514],{"class":3666},[3483,5347,5348],{"class":4517}," Main",[3483,5350,5351],{"class":3673},"()",[3483,5353,5354],{"class":3673}," ->",[3483,5356,5357],{"class":3666}," int",[3483,5359,3683],{"class":3673},[3483,5361,5363],{"class":3485,"line":5362},83,[3483,5364,5365],{"class":4305},"    \u002F\u002F An odd count: one value sits exactly in the middle.\n",[3483,5367,5369,5371,5373,5375,5377,5379,5382,5384,5386,5388,5391,5393,5396,5398,5401,5403,5405,5407,5410,5412,5415,5417,5420,5422,5424,5426,5429,5431,5434,5436,5439,5441,5443,5445,5448,5450,5452,5454,5457,5459,5461,5463,5465,5467,5469],{"class":3485,"line":5368},84,[3483,5370,4929],{"class":3666},[3483,5372,4320],{"class":3669},[3483,5374,4249],{"class":3673},[3483,5376,3834],{"class":3833},[3483,5378,3713],{"class":3673},[3483,5380,5381],{"class":3763},"9",[3483,5383,4259],{"class":3673},[3483,5385,4262],{"class":3691},[3483,5387,4265],{"class":3673},[3483,5389,5390],{"class":3763},"12",[3483,5392,3435],{"class":3673},[3483,5394,5395],{"class":3763},"5",[3483,5397,3707],{"class":3673},[3483,5399,5400],{"class":3763}," 9",[3483,5402,3435],{"class":3673},[3483,5404,3861],{"class":3763},[3483,5406,3707],{"class":3673},[3483,5408,5409],{"class":3763}," 15",[3483,5411,3435],{"class":3673},[3483,5413,5414],{"class":3763},"25",[3483,5416,3707],{"class":3673},[3483,5418,5419],{"class":3763}," 11",[3483,5421,3435],{"class":3673},[3483,5423,3861],{"class":3763},[3483,5425,3707],{"class":3673},[3483,5427,5428],{"class":3763}," 8",[3483,5430,3435],{"class":3673},[3483,5432,5433],{"class":3763},"75",[3483,5435,3707],{"class":3673},[3483,5437,5438],{"class":3763}," 14",[3483,5440,3435],{"class":3673},[3483,5442,3861],{"class":3763},[3483,5444,3707],{"class":3673},[3483,5446,5447],{"class":3763}," 10",[3483,5449,3435],{"class":3673},[3483,5451,5395],{"class":3763},[3483,5453,3707],{"class":3673},[3483,5455,5456],{"class":3763}," 13",[3483,5458,3435],{"class":3673},[3483,5460,5414],{"class":3763},[3483,5462,3707],{"class":3673},[3483,5464,5419],{"class":3763},[3483,5466,3435],{"class":3673},[3483,5468,5433],{"class":3763},[3483,5470,4133],{"class":3673},[3483,5472,5474,5477,5479,5482,5484,5486,5488,5490],{"class":3485,"line":5473},85,[3483,5475,5476],{"class":3669},"    Summarize",[3483,5478,3674],{"class":3673},[3483,5480,5481],{"class":3703},"\"nine readings\"",[3483,5483,3707],{"class":3673},[3483,5485,4320],{"class":3669},[3483,5487,3713],{"class":3673},[3483,5489,4292],{"class":3691},[3483,5491,4100],{"class":3673},[3483,5493,5495],{"class":3485,"line":5494},86,[3483,5496,4300],{"emptyLinePlaceholder":4299},[3483,5498,5500],{"class":3485,"line":5499},87,[3483,5501,5502],{"class":4305},"    \u002F\u002F An even count with repeated values. Duplicates are ordinary data: each one counts.\n",[3483,5504,5506,5508,5511,5513,5515,5517,5520,5522,5524,5526,5529,5531,5533,5535,5538,5540,5542,5544,5547,5549,5551,5553,5555,5557,5559,5561,5563,5565,5567,5569,5571,5573,5575],{"class":3485,"line":5505},88,[3483,5507,4929],{"class":3666},[3483,5509,5510],{"class":3669}," scores",[3483,5512,4249],{"class":3673},[3483,5514,3834],{"class":3833},[3483,5516,3713],{"class":3673},[3483,5518,5519],{"class":3763},"6",[3483,5521,4259],{"class":3673},[3483,5523,4262],{"class":3691},[3483,5525,4265],{"class":3673},[3483,5527,5528],{"class":3763},"4",[3483,5530,3435],{"class":3673},[3483,5532,3861],{"class":3763},[3483,5534,3707],{"class":3673},[3483,5536,5537],{"class":3763}," 7",[3483,5539,3435],{"class":3673},[3483,5541,3861],{"class":3763},[3483,5543,3707],{"class":3673},[3483,5545,5546],{"class":3763}," 4",[3483,5548,3435],{"class":3673},[3483,5550,3861],{"class":3763},[3483,5552,3707],{"class":3673},[3483,5554,4077],{"class":3763},[3483,5556,3435],{"class":3673},[3483,5558,3861],{"class":3763},[3483,5560,3707],{"class":3673},[3483,5562,5537],{"class":3763},[3483,5564,3435],{"class":3673},[3483,5566,3861],{"class":3763},[3483,5568,3707],{"class":3673},[3483,5570,5537],{"class":3763},[3483,5572,3435],{"class":3673},[3483,5574,3861],{"class":3763},[3483,5576,4133],{"class":3673},[3483,5578,5580,5582,5584,5587,5589,5591,5593,5595],{"class":3485,"line":5579},89,[3483,5581,5476],{"class":3669},[3483,5583,3674],{"class":3673},[3483,5585,5586],{"class":3703},"\"six scores with repeats\"",[3483,5588,3707],{"class":3673},[3483,5590,5510],{"class":3669},[3483,5592,3713],{"class":3673},[3483,5594,4292],{"class":3691},[3483,5596,4100],{"class":3673},[3483,5598,5600],{"class":3485,"line":5599},90,[3483,5601,4300],{"emptyLinePlaceholder":4299},[3483,5603,5605],{"class":3485,"line":5604},91,[3483,5606,5607],{"class":4305},"    \u002F\u002F One value: no spread at all, and no sample variance.\n",[3483,5609,5611,5613,5615,5617,5619,5621,5623,5625,5627,5629,5631,5633,5635],{"class":3485,"line":5610},92,[3483,5612,4929],{"class":3666},[3483,5614,4246],{"class":3669},[3483,5616,4249],{"class":3673},[3483,5618,3834],{"class":3833},[3483,5620,3713],{"class":3673},[3483,5622,4256],{"class":3763},[3483,5624,4259],{"class":3673},[3483,5626,4262],{"class":3691},[3483,5628,4265],{"class":3673},[3483,5630,4268],{"class":3763},[3483,5632,3435],{"class":3673},[3483,5634,3861],{"class":3763},[3483,5636,4133],{"class":3673},[3483,5638,5640,5642,5644,5646,5648,5650,5652,5654],{"class":3485,"line":5639},93,[3483,5641,5476],{"class":3669},[3483,5643,3674],{"class":3673},[3483,5645,4283],{"class":3703},[3483,5647,3707],{"class":3673},[3483,5649,4246],{"class":3669},[3483,5651,3713],{"class":3673},[3483,5653,4292],{"class":3691},[3483,5655,4100],{"class":3673},[3483,5657,5659],{"class":3485,"line":5658},94,[3483,5660,4300],{"emptyLinePlaceholder":4299},[3483,5662,5664],{"class":3485,"line":5663},95,[3483,5665,5666],{"class":4305},"    \u002F\u002F An empty view of an array, so there is nothing to summarise.\n",[3483,5668,5670,5672,5674,5676,5678,5680,5682,5684,5686],{"class":3485,"line":5669},96,[3483,5671,5476],{"class":3669},[3483,5673,3674],{"class":3673},[3483,5675,4315],{"class":3703},[3483,5677,3707],{"class":3673},[3483,5679,4320],{"class":3669},[3483,5681,3713],{"class":3673},[3483,5683,4292],{"class":3691},[3483,5685,3861],{"class":3763},[3483,5687,4100],{"class":3673},[3483,5689,5691,5693,5695],{"class":3485,"line":5690},97,[3483,5692,3791],{"class":3666},[3483,5694,3764],{"class":3763},[3483,5696,3794],{"class":3673},[3483,5698,5700],{"class":3485,"line":5699},98,[3483,5701,3742],{"class":3673},[3411,5703,5704,5705,5707,5708,5711,5712,3432,5714,5716,5717,3435],{},"Besides ",[3447,5706,2711],{},", its ",[3447,5709,5710],{},"Rux.toml"," lists ",[3447,5713,888],{},[3447,5715,2819],{}," under ",[3447,5718,5719],{},"[Dependencies]",[3459,5721,5723],{"id":5722},"run-it","Run it",[3474,5725,5729],{"className":5726,"code":5727,"language":5728,"meta":3479,"style":3479},"language-sh shiki shiki-themes catppuccin-latte catppuccin-latte catppuccin-mocha","cd Examples\u002FProjects\u002FStatistics\nrux run\n","sh",[3447,5730,5731,5740],{"__ignoreMap":3479},[3483,5732,5733,5737],{"class":3485,"line":3486},[3483,5734,5736],{"class":5735},"s0rF6","cd",[3483,5738,5739],{"class":3703}," Examples\u002FProjects\u002FStatistics\n",[3483,5741,5742,5744],{"class":3485,"line":3492},[3483,5743,3659],{"class":4517},[3483,5745,5746],{"class":3703}," run\n",[3474,5748,5753],{"className":5749,"code":5751,"language":5752},[5750],"language-text","nine readings\n  values     12.5 9.0 15.25 11.0 8.75 14.0 10.5 13.25 11.75\n  count      9\n  smallest   8.75\n  largest    15.25\n  mean       11.7778\n  population variance 4.3117, deviation 2.0765\n  sample     variance 4.8507, deviation 2.2024\n  median     11.75 (the middle value)\n\nsix scores with repeats\n  values     4.0 7.0 4.0 1.0 7.0 7.0\n  count      6\n  smallest   1.0\n  largest    7.0\n  mean       5.0000\n  population variance 5.0000, deviation 2.2361\n  sample     variance 6.0000, deviation 2.4495\n  median     5.5 (between 4.0 and 7.0)\n\na single value\n  values     42.0\n  count      1\n  smallest   42.0\n  largest    42.0\n  mean       42.0000\n  population variance 0.0000, deviation 0.0000\n  sample     variance undefined for one value\n  median     42.0 (the middle value)\n\nno values\n  values\n  count      0\n  smallest   none\n  largest    none\n  mean, variance and median need at least one value\n","text",[3447,5754,5751],{"__ignoreMap":3479},[3459,5756,5758],{"id":5757},"common-mistakes","Common mistakes",[5760,5761,5762],"warning",{},[3411,5763,5764,5767,5770,5772,5773,5775,5776,5778,5779,5782,5783,5785,5786,5788,5789,5792],{},[3414,5765,5766],{},"Passing data that cannot be sorted.",[5768,5769],"br",{},[3447,5771,3468],{}," sorts, so its parameter is ",[3447,5774,4214],{},". An array declared with ",[3447,5777,3819],{}," gives only a read-only view, and the call fails: ",[3447,5780,5781],{},"error: argument 2 to 'Summarize' has type 'float64[..]', but parameter 'values' requires 'var float64[..]'",". Dropping the ",[3447,5784,3876],{}," from the parameter just moves the problem to ",[3447,5787,3425],{},", which needs a ",[3447,5790,5791],{},"var T[..]"," too.",[5760,5794,5795],{},[3411,5796,5797,5800,5802,5803,5806],{},[3414,5798,5799],{},"Taking the median before sorting.",[5768,5801],{},"\nWithout ",[3447,5804,5805],{},"Sort(values);"," the program still runs, but the \"median\" is whatever sits in the middle of the unsorted data: 8.75 for the nine readings instead of 11.75.",[5760,5808,5809],{},[3411,5810,5811,5814,5816,5817,5820,5821,5824,5825,4527,5828,5831],{},[3414,5812,5813],{},"Forgetting the empty case.",[5768,5815],{},"\nRemove the ",[3447,5818,5819],{},"count == 0"," check and the empty summary prints a mean of ",[3447,5822,5823],{},"NaN"," — zero divided by zero — and then stops with ",[3447,5826,5827],{},"Panic: index out of range",[3447,5829,5830],{},"middle - 1"," on a count of 0 is not −1 but a huge unsigned number.",[3459,5833,5835],{"id":5834},"try-it-yourself","Try it yourself",[5837,5838,5839,5847,5853,5856],"ol",{},[5840,5841,5842,5843,5846],"li",{},"Add the ",[3414,5844,5845],{},"range"," (largest minus smallest) to the summary. Where must it go so that it still works for an empty slice?",[5840,5848,5842,5849,5852],{},[3414,5850,5851],{},"mode",", the most frequent value. Because the slice is sorted by then, equal values sit side by side, so one pass that counts runs is enough.",[5840,5854,5855],{},"Print the values again after the median, to see that the caller's array really has been sorted.",[5840,5857,5858,5859,5861,5862,5865],{},"Summarise a copy instead: make ",[3447,5860,3468],{}," take a read-only ",[3447,5863,5864],{},"float64[..]"," and sort a copy of the data in a local array, so the caller's data keeps its order. What limits the size of that local array?",[3459,5867,5869],{"id":5868},"learn-more","Learn more",[5871,5872,5873,5881,5888,5893],"ul",{},[5840,5874,5875,3422,5877,3432,5879],{},[3419,5876,3428],{"href":908},[3419,5878,3431],{"href":912},[3419,5880,3425],{"href":896},[5840,5882,5883,5885,5886],{},[3419,5884,3421],{"href":311}," — why sorting needs ",[3447,5887,5791],{},[5840,5889,5890,5892],{},[3419,5891,2933],{"href":2934}," in the API reference",[5840,5894,5895,5896,3422,5899,3422,5902,3432,5905,5908],{},"Next projects: ",[3419,5897,5898],{"href":1189},"Guess",[3419,5900,5901],{"href":1193},"Age",[3419,5903,5904],{"href":1197},"Password",[3419,5906,5907],{"href":1201},"Launch",", the checkpoints for Utilities",[5910,5911,5912],"style",{},"html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span 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.s0rF6{--shiki-light:#D20F39;--shiki-light-font-style:italic;--shiki-default:#D20F39;--shiki-default-font-style:italic;--shiki-dark:#F38BA8;--shiki-dark-font-style:italic}",{"title":3479,"searchDepth":3492,"depth":3492,"links":5914},[5915,5916,5917,5918,5919,5920,5921,5922,5923,5924,5925],{"id":3461,"depth":3492,"text":3462},{"id":3638,"depth":3492,"text":3639},{"id":3801,"depth":3492,"text":3802},{"id":3969,"depth":3492,"text":3970},{"id":4022,"depth":3492,"text":4023},{"id":4221,"depth":3492,"text":4222},{"id":4331,"depth":3492,"text":4332},{"id":5722,"depth":3492,"text":5723},{"id":5757,"depth":3492,"text":5758},{"id":5834,"depth":3492,"text":5835},{"id":5868,"depth":3492,"text":5869},"Summarise slices of numbers — count, extremes, mean, population and sample variance, and median — including an even count with repeated values, a single value and no values at all.","md",{"number":5929,"source":5930,"requires":5931},"25.10","Projects\u002FStatistics",[5932,5933,5934,5935,5936],"writable-slice","sort","min-max","fold","math",{},{"title":1184},{"title":3398,"description":5926},"F844KTQkaaVNL3yTJQdibRaYlVNBSNHDQu_ivK-58-I",[5942,5944],{"title":1180,"path":1181,"stem":1182,"description":5943,"children":-1},"Keep a shop's stock as structs in a vector and apply a day of orders, refusing the ones that cannot be carried out.",{"title":1188,"path":1189,"stem":1190,"description":5945,"children":-1},"A number guessing game: find a number from 1 to 100 in seven guesses, with a hint after each one.",1791225606524]