3.7. Standardizing data#

Some data are recorded in naturally meaningful units; examples are

  • height of adults in cm

  • temperature in \(^{\circ}C\)

In other cases, units may be hard to interpret because we don’t have a sense of what a typical score is, based on general knowledge

  • scores on an IQ test marked out of 180

  • height of 6-year-olds in cm

A further problem is quantifying how unusual a data value is when values are presented as different units

  • High school grades from different countries or systems (A-levels vs IB vs Abitur vs…..)

In all cases it can be useful to present data in standard units.

Two common ways of doing this are:

  • Convert data to Z-scores

  • Convert data to quantiles

In this section we will review both these approaches.

Here is a video about standardizing data using centiles

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Here is a video about standardizing data using Z-Scores

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3.7.1. Set up Python Libraries#

As usual you will need to run this code block to import the relevant Python libraries

# Set-up Python libraries - you need to run this but you don't need to change it
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
import pandas as pd
import seaborn as sns
sns.set_theme(style='white')
import statsmodels.api as sm
import statsmodels.formula.api as smf

3.7.2. Import a dataset to work with#

Let’s look at a fictional dataset containing some body measurements for 50 individuals

data=pd.read_csv('https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook_2024/main/data/BodyData.csv')
display(data)
ID sex height weight age
0 101708 M 161 64.8 35
1 101946 F 165 68.1 42
2 108449 F 175 76.6 31
3 108796 M 180 81.0 31
4 113449 F 179 80.1 31
5 114688 M 172 74.0 42
6 119187 F 148 54.8 45
7 120679 F 160 64.0 44
8 120735 F 188 88.4 32
9 124269 F 172 74.0 29
10 124713 M 175 76.6 26
11 127076 M 180 81.0 28
12 131626 M 162 65.6 35
13 132218 M 170 72.3 29
14 132609 F 172 74.0 41
15 134660 F 159 63.2 34
16 135195 M 169 71.4 42
17 140073 F 168 70.6 34
18 140114 M 195 95.1 41
19 145185 F 157 61.6 45
20 146279 F 180 81.0 30
21 146519 F 172 74.0 34
22 151451 F 171 73.1 37
23 152597 M 172 74.0 27
24 154672 M 167 69.7 39
25 155594 F 165 68.1 25
26 158165 M 175 76.6 45
27 159457 F 176 77.4 36
28 162323 M 173 74.8 31
29 166948 M 174 75.7 28
30 168411 M 175 76.6 29
31 168574 F 163 66.4 30
32 169209 F 159 63.2 45
33 171236 F 164 67.2 34
34 172289 M 181 81.9 27
35 173925 M 189 89.3 25
36 176598 F 169 71.4 37
37 177002 F 180 81.0 36
38 178659 M 181 81.9 26
39 180992 F 177 78.3 31
40 183304 F 176 77.4 30
41 184706 M 183 83.7 40
42 185138 M 169 71.4 28
43 185223 F 170 72.3 41
44 186041 M 175 76.6 25
45 186887 M 154 59.3 26
46 187016 M 161 64.8 32
47 198157 M 180 81.0 33
48 199112 M 172 74.0 33
49 199614 F 164 67.2 31

3.7.3. Z score#

The Z-score tells us how many standard deviations above or below the mean of the distribution a given value lies.

Let’s convert our weights to Z-scores. We will need to know the mean and standard deviation of weight:

print(data.weight.mean())
print(data.weight.std())
73.73
7.891438140058334

We can then calculate a Z-score for each person’s weight.

  • Someone whose weight is exactly on the mean (74kg) will have a Z-score of 0.

  • Someone whose weight is one standard deviation below the mean (65kg) will have a Z-score of -1 etc

# Create a new column and put the calcualted z-scores in it
data['WeightZ'] = (data.weight - data.weight.mean())/data.weight.std()
data
ID sex height weight age WeightZ
0 101708 M 161 64.8 35 -1.131606
1 101946 F 165 68.1 42 -0.713431
2 108449 F 175 76.6 31 0.363685
3 108796 M 180 81.0 31 0.921252
4 113449 F 179 80.1 31 0.807204
5 114688 M 172 74.0 42 0.034214
6 119187 F 148 54.8 45 -2.398802
7 120679 F 160 64.0 44 -1.232982
8 120735 F 188 88.4 32 1.858977
9 124269 F 172 74.0 29 0.034214
10 124713 M 175 76.6 26 0.363685
11 127076 M 180 81.0 28 0.921252
12 131626 M 162 65.6 35 -1.030230
13 132218 M 170 72.3 29 -0.181209
14 132609 F 172 74.0 41 0.034214
15 134660 F 159 63.2 34 -1.334358
16 135195 M 169 71.4 42 -0.295257
17 140073 F 168 70.6 34 -0.396632
18 140114 M 195 95.1 41 2.707998
19 145185 F 157 61.6 45 -1.537109
20 146279 F 180 81.0 30 0.921252
21 146519 F 172 74.0 34 0.034214
22 151451 F 171 73.1 37 -0.079833
23 152597 M 172 74.0 27 0.034214
24 154672 M 167 69.7 39 -0.510680
25 155594 F 165 68.1 25 -0.713431
26 158165 M 175 76.6 45 0.363685
27 159457 F 176 77.4 36 0.465061
28 162323 M 173 74.8 31 0.135590
29 166948 M 174 75.7 28 0.249638
30 168411 M 175 76.6 29 0.363685
31 168574 F 163 66.4 30 -0.928855
32 169209 F 159 63.2 45 -1.334358
33 171236 F 164 67.2 34 -0.827479
34 172289 M 181 81.9 27 1.035299
35 173925 M 189 89.3 25 1.973024
36 176598 F 169 71.4 37 -0.295257
37 177002 F 180 81.0 36 0.921252
38 178659 M 181 81.9 26 1.035299
39 180992 F 177 78.3 31 0.579109
40 183304 F 176 77.4 30 0.465061
41 184706 M 183 83.7 40 1.263395
42 185138 M 169 71.4 28 -0.295257
43 185223 F 170 72.3 41 -0.181209
44 186041 M 175 76.6 25 0.363685
45 186887 M 154 59.3 26 -1.828564
46 187016 M 161 64.8 32 -1.131606
47 198157 M 180 81.0 33 0.921252
48 199112 M 172 74.0 33 0.034214
49 199614 F 164 67.2 31 -0.827479

Look down the table for some heavy and light people. Do their z-scores look like you would expect?

Disadvantages of the Z score#

Z score tells us how many standard deviations above or below the mean a datapoint lies.

We can use some hand ‘rules of thumb’ to know how unusual a Z-score is, as long as the data distribution is approximately normal: * Don’t worry if you don’t know what the Normal distribution is yet - you will learn about this in detail later in the course

https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook_2024/main/images/MT_wk3_ZscoreRuleOfThumb.png

The Z-score does have a couple of disadvantages:

  • it is only really meaningful for symmetrical data distributions (especially the Normal distribution) - for skewed distributions, there will be momre datapoints with a Z-score of, say, +2, than -2

https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook_2024/main/images/MT_wk3_ZscoreSkew.png

Additionally, the Z-score is not easily understood by non statistically trained people

It is therefore sometimes more meaningful to standardize data by presenting them as quantiles

3.7.4. Quantiles#

Quantiles (or centiles) tell us what proportion of data points are expected to exceed a certain value. This is easy to interpret.

For example, say my six year old daughter is 125cm tall, would you say she is tall for her age? You probably have no idea - this is in contrast to adult heights where people might have a sense of the distribution due to general knowledge (eg 150cm is small and 180cm is tall)

In fact, a a 6 year old with height 125cm lies on the 95th centile, which means they are taller than 95% of children the same age (will definitley look tall in the playground).

To calculate a given quantile of a dataset we use df.quantile(), eg

# find the 90th centile for height in out dataframe
data.height.quantile(q=0.9) # get 90th centile
181.0

The 90th centile is 181cm, ie 10% of people are taller than 181cm.

Adding quantiles to the table can be done using df.qcut(), which categorizes the data into quantiles. For example, I can produe a table saying which decile each person’s weight falls into as follows:

  • Deciles are 10ths, in the same way that centiles are 100ths

    • if someone’s weight in the 0th decile, that means their weight is in the bottom 10% of the sample

    • if someone’s weight is in the 9th decile, it mmeans they are in teh top 10% ot the sample (heavier than 90% of people)

data['weightQ'] = pd.qcut(data.weight, 10, labels = False) 
data
ID sex height weight age WeightZ weightQ
0 101708 M 161 64.8 35 -1.131606 1
1 101946 F 165 68.1 42 -0.713431 2
2 108449 F 175 76.6 31 0.363685 6
3 108796 M 180 81.0 31 0.921252 7
4 113449 F 179 80.1 31 0.807204 7
5 114688 M 172 74.0 42 0.034214 4
6 119187 F 148 54.8 45 -2.398802 0
7 120679 F 160 64.0 44 -1.232982 1
8 120735 F 188 88.4 32 1.858977 9
9 124269 F 172 74.0 29 0.034214 4
10 124713 M 175 76.6 26 0.363685 6
11 127076 M 180 81.0 28 0.921252 7
12 131626 M 162 65.6 35 -1.030230 1
13 132218 M 170 72.3 29 -0.181209 3
14 132609 F 172 74.0 41 0.034214 4
15 134660 F 159 63.2 34 -1.334358 0
16 135195 M 169 71.4 42 -0.295257 3
17 140073 F 168 70.6 34 -0.396632 3
18 140114 M 195 95.1 41 2.707998 9
19 145185 F 157 61.6 45 -1.537109 0
20 146279 F 180 81.0 30 0.921252 7
21 146519 F 172 74.0 34 0.034214 4
22 151451 F 171 73.1 37 -0.079833 4
23 152597 M 172 74.0 27 0.034214 4
24 154672 M 167 69.7 39 -0.510680 2
25 155594 F 165 68.1 25 -0.713431 2
26 158165 M 175 76.6 45 0.363685 6
27 159457 F 176 77.4 36 0.465061 7
28 162323 M 173 74.8 31 0.135590 5
29 166948 M 174 75.7 28 0.249638 5
30 168411 M 175 76.6 29 0.363685 6
31 168574 F 163 66.4 30 -0.928855 1
32 169209 F 159 63.2 45 -1.334358 0
33 171236 F 164 67.2 34 -0.827479 2
34 172289 M 181 81.9 27 1.035299 8
35 173925 M 189 89.3 25 1.973024 9
36 176598 F 169 71.4 37 -0.295257 3
37 177002 F 180 81.0 36 0.921252 7
38 178659 M 181 81.9 26 1.035299 8
39 180992 F 177 78.3 31 0.579109 7
40 183304 F 176 77.4 30 0.465061 7
41 184706 M 183 83.7 40 1.263395 9
42 185138 M 169 71.4 28 -0.295257 3
43 185223 F 170 72.3 41 -0.181209 3
44 186041 M 175 76.6 25 0.363685 6
45 186887 M 154 59.3 26 -1.828564 0
46 187016 M 161 64.8 32 -1.131606 1
47 198157 M 180 81.0 33 0.921252 7
48 199112 M 172 74.0 33 0.034214 4
49 199614 F 164 67.2 31 -0.827479 2

NOTE this is a bit fiddly as df.qcut won’t create empty bins. Since this dataset is quite small, we can’t create one bin for each centile as naturally some will be empty (as there are less than 100 datapoints)