As a pastime, I’m engaged on a SwiftUI app on the facet. It permits me to maintain observe of peak and weight of my daughters and plot them on charts that permit me to see how “regular” my offspring are growing.

I’ve shied away from statistics at college, so it took me so time to analysis just a few issues to resolve a difficulty I used to be having. Let me share how I labored in direction of an answer to this statistical drawback. Could you discover it as instructive as I did.

Notice: Should you discover any error of thought or truth on this article, please let me know on Twitter, in order that I can perceive what prompted it.

Let me first provide you with some background as to what I’ve completed earlier than right this moment, so that you simply perceive my statistical query.

### Setup

The World Well being Group publishes tables that give the percentiles for size/peak from delivery to 2 years, to 5 years and to 19 years. Till two years of age the measurement is to be carried out with the toddler on its again, and referred to as “size”. Past two years we measure standing up after which it’s referred to as “peak”. That’s why there’s a slight break within the printed values at two years.

I additionally compiled my ladies heights in a Numbers sheet which I fed from paediatrician visits initially and later by often marking their peak on a poster in the back of their bed room door.

To get began I hard-coded the heights such:

```
import Basis
struct ChildData
{
let days: Int
let peak: Double
}
let elise = [ChildData(days: 0, height: 50),
ChildData(days: 6, height: 50),
ChildData(days: 49, height: 60),
ChildData(days: 97, height: 64),
ChildData(days: 244, height: 73.5),
ChildData(days: 370, height: 78.5),
ChildData(days: 779, height: 87.7),
ChildData(days: 851, height: 90),
ChildData(days: 997, height: 95),
ChildData(days: 1178, height: 97.5),
ChildData(days: 1339, height: 100),
ChildData(days: 1367, height: 101),
ChildData(days: 1464, height: 103.0),
ChildData(days: 1472, height: 103.4),
ChildData(days: 1544, height: 105),
ChildData(days: 1562, height: 105.2)
]
let erika = [ChildData(days: 0, height: 47),
ChildData(days: 7, height: 48),
ChildData(days: 44, height: 54),
ChildData(days: 119, height: 60.5),
ChildData(days: 256, height: 68.5),
ChildData(days: 368, height: 72.5),
ChildData(days: 529, height: 80),
ChildData(days: 662, height: 82),
ChildData(days: 704, height: 84),
ChildData(days: 734, height: 85),
ChildData(days: 752, height: 86),
]
```

The WHO outlined one month as 30.4375 days and so I used to be in a position to have these values be plotted on a SwiftUI chart. The vertical traces you see on the chart are months with bolder traces representing full years. You may as well discover the small step on the second yr finish.

It’s nonetheless lacking some kind of labelling, however you possibly can already see that my older daughter Elise (blue) was on the taller facet throughout her first two years, whereas the second-born Erika (purple) was fairly near the “center of the highway”.

This chart offers you an eye-eye overview of the place on the highway my daughters are, however I needed to have the ability to put your finger down on each place and have a pop up let you know the precise percentile worth.

### The Information Dilemma

A percentile worth is mainly giving the knowledge what number of p.c of kids are shorter than your little one. So in case your child is on the seventy fifth percentile, then seventy fifth of kids are shorter than it. The shades of inexperienced on the chart symbolize the steps within the uncooked information supplied by the WHO.

Thery provide you with `P01, P1, P3, P5, P10, P15, P25, P50, P75, P85, P90, P95, P97, P99, P999.`

`P01`

is the 0.1th percentile, `P999`

is the 99.ninth percentile. On the extremes the percentiles are very shut collectively, however within the center there’s a enormous bounce from 25 to 50 to 75.

I needed to indicate percentile values at these arbitrary instances which can be not less than full integers. i.e. say forty seventh percentile as an alternative of “between 25 and 50” and possibly present this place with a coloured line on the distribution curve these percentile values symbolize.

It seems, these peak values are “usually distributed”, on a curve that appears a bit like a bell, thus the time period “bell curve”. To me as a programmer, I’d say that I perceive {that a} a type a knowledge compression the place you solely must to know the imply worth and the usual deviation and from that you would be able to draw the curve, versus interpolating between the person percentile values.

The second – smaller – challenge is that WHO offers information for full months solely. To find out the conventional distribution curve for arbitrary instances in between the months we have to interpolate between the month information earlier than and after the arbitrary worth.

With these questions I turned to Stack Overflow and Math Stack Change hoping that anyone might assist out me statistics noob. Right here’s what I posted…

### The Drawback

Given the size percentiles information the WHO has printed for women. That’s size in cm at for sure months. e.g. at delivery the 50% percentile is 49.1 cm.

```
Month L M S SD P01 P1 P3 P5 P10 P15 P25 P50 P75 P85 P90 P95 P97 P99 P999
0 1 49.1477 0.0379 1.8627 43.4 44.8 45.6 46.1 46.8 47.2 47.9 49.1 50.4 51.1 51.5 52.2 52.7 53.5 54.9
1 1 53.6872 0.0364 1.9542 47.6 49.1 50 50.5 51.2 51.7 52.4 53.7 55 55.7 56.2 56.9 57.4 58.2 59.7
2 1 57.0673 0.03568 2.0362 50.8 52.3 53.2 53.7 54.5 55 55.7 57.1 58.4 59.2 59.7 60.4 60.9 61.8 63.4
3 1 59.8029 0.0352 2.1051 53.3 54.9 55.8 56.3 57.1 57.6 58.4 59.8 61.2 62 62.5 63.3 63.8 64.7 66.3
```

P01 is the 0.1% percentile, P1 the 1% percentile and P50 is the 50% percentile.

Say, I’ve a sure (doubtlessly fractional) month, say 2.3 months. (a peak measurement can be carried out at a sure variety of days after delivery and you’ll divide that by 30.4375 to get a fractional month)

How would I am going about approximating the percentile for a selected peak at a fraction month? i.e. as an alternative of simply seeing it “subsequent to P50”, to say, effectively that’s about “P62”

One method I considered can be to do a linear interpolation, first between month 2 and month 3 between all mounted percentile values. After which do a linear interpolation between P50 and P75 (or these two percentiles for which there’s information) values of these time-interpolated values.

What I concern is that as a result of this can be a bell curve the linear values close to the center is likely to be too far off to be helpful.

So I’m pondering, is there some components, e.g. a quad curve that you may use with the mounted percentile values after which get a precise worth on this curve for a given measurement?

This bell curve is a traditional distribution, and I suppose there’s a components by which you may get values on the curve. The temporal interpolation can most likely nonetheless be carried out linear with out inflicting a lot distortion.

### My Answer

I did get some responses starting from ineffective to a degree the place they is likely to be right, however to me as a math outsider they didn’t assist me obtain my aim. So I got down to analysis how one can obtain the end result myself.

I labored via the query primarily based on two examples, specifically my two daughters.

ELISE at 49 days divide by 30.4375 = 1.61 months 60 cm

In order that’s between month 1 and month 2:

Month P01 P1 P3 P5 P10 P15 P25 P50 P75 P85 P90 P95 P97 P99 P999 1 47.6 49.1 50 50.5 51.2 51.7 52.4 53.7 55 55.7 56.2 56.9 57.4 58.2 59.7 2 50.8 52.3 53.2 53.7 54.5 55 55.7 57.1 58.4 59.2 59.7 60.4 60.9 61.8 63.4

Subtract the decrease month: 1.61 – 1 = 0.61. So the worth is 61% the best way to month 2. I’d get a percentile row for this by linear interpolation. For every percentile I can interpolate values from the month row earlier than and after it.

```
// e.g. for P01
p1 = 47.6
p2 = 50.8
p1 * (1.0 - 0.61) + p2 * (0.61) = 18.564 + 30.988 = 49.552
```

I did that in Numbers to get the values for all percentile columns.

```
Month P01 P1 P3 P5 P10 P15 P25 P50 P75 P85 P90 P95 P97 P99 P999
1.6 49.552 51.052 51.952 52.452 53.213 53.713 54.413 55.774 57.074 57.835 58.335 59.035 59.535 60.396 61.957
```

First, I attempted the linear interpolation:

60 cm is between 59,535 (P97) and 60,396 (P99).

0.465 away from the decrease, 0.396 away from the upper worth.

0.465 is 54% of the space between them (0,861)

```
(1-0.54) * 97 + 0.54 * 99 = 44.62 + 53.46 = 98,08
// rounded P98
```

Seems that this can be a *dangerous instance*.

On the extremes the percentiles are very carefully spaced in order that linear interpolation would give related outcomes. Linear interpolation within the center can be too inaccurate.

Let’s do a greater instance. This time with my second daughter.

ERIKA at 119 days divide by 30.4375 = 3.91 months 60.5 cm

We interpolate between month 3 and month 4:

Month P01 P1 P3 P5 P10 P15 P25 P50 P75 P85 P90 P95 P97 P99 P999 3 53.3 54.9 55.8 56.3 57.1 57.6 58.4 59.8 61.2 62.0 62.5 63.3 63.8 64.7 66.3 4 55.4 57.1 58.0 58.5 59.3 59.8 60.6 62.1 63.5 64.3 64.9 65.7 66.2 67.1 68.8 3.91 55.211 56.902 57.802 58.302 59.102 59.602 60.402 61.893 63.293 64.093 64.684 65.484 65.984 66.884 68.575

Once more, let’s strive with linear interpolation:

60.5 cm is between 60.402 (P25) and 61.893 (P50)

0.098 of the space 1.491 = 6.6%

P = 25 * (1-0.066) + 50 * 0.066 = 23.35 + 3.3 = 26.65 // rounds toP27

To check that to approximating it on a bell curve, I used an on-line calculator/plotter. This wanted a imply and an ordinary deviation, which I feel I discovered on the percentile desk left-most columns. However I additionally must interpolate these for month 3.91:

Month L M S SD 3 1.0 59.8029 0.0352 2.1051 4 1.0 62.0899 0.03486 2.1645 3.91 1.0 61.88407 0.0348906 2.159154

I don’t know what L and S imply, however M most likely means MEAN and SD most likely means Customary Deviation`.`

Plugging these into the web plotter…

μ = 61.88407

σ = 2.159154

x = 60.5

The net plotter offers me a results of P(X < x) = 0.26075, rounded **P26**

That is far sufficient from the **P27** I arrived at by linear interpolation, warranting a extra correct method.

### Z-Scores Tables

Looking round, I discovered that when you can convert a size worth right into a z-score you possibly can then lookup the percentile in a desk. I additionally discovered this nice clarification of Z-Scores.

Z-Rating is the variety of commonplace deviation from the imply {that a} sure information level is.

So I’m attempting to realize the identical end result as above with the components:

(x - M) / SD

(60.5 - 61.88407) / 2.159154 = -0.651

Then I used to be in a position to convert that right into a percentile by consulting a z-score desk.

Wanting up `-0.6`

on the left facet vertically after which `0.05`

horizontally I get to `0.25785`

– In order that rounds to be additionally **P26**, though I get an uneasy feeling that it’s ever so barely lower than the worth spewed out from the calculator.

### How to try this in Swift?

Granted that it might be *easy sufficient* to implement such a percentile lookup desk in Swift, however the feeling that I can get a extra correct end result coupled with much less work pushed me to search around for a Swift bundle.

Certainly, Sigma Swift Statistics appears to supply the wanted statistics operate “regular distribution”, described as:

Returns the conventional distribution for the given values of x, μ and σ. The returned worth is the world beneath the conventional curve to the left of the worth x.

I couldn’t discover something talked about percentile as end result, however I added the Swift bundle and I attempted it out for the second instance, to see what end result I’d get for this worth between P25 and P50:

let y = Sigma.normalDistribution(x: 60, μ: 55.749061, σ: 2.00422) // end result 0.2607534748851712

That appears very shut sufficient to **P26**. It’s totally different than the worth from the z-tables, `0.25785` nevertheless it rounds to the identical integer percentile worth.

For the primary instance, between P97 and P99, we additionally get inside rounding distance of **P98**.

let y = Sigma.normalDistribution(x: 60, μ: 55.749061, σ: 2.00422) // end result 0.9830388548349042

As a facet word, I discovered it pleasant to see using greek letters for the parameters, a function potential as a result of Swifts Unicode help.

### Conclusion

Math and statistics had been the explanation why I aborted my college diploma in laptop science. I couldn’t see how these would have benefitted me “in actual life” as a programmer.

Now – many many years later – I often discover {that a} bit extra data in these issues would permit me to grasp such uncommon situations extra rapidly. Fortunately, my web looking expertise could make up for what I lack in educational data.

I appear to have the components assembled to begin engaged on this regular distribution chart giving interpolated percentile values for particular days between the month boundaries. I’ll give an replace when I’ve constructed that, if you’re .

Additionally printed on Medium.