How Would The Correlations Change If We Normalized The Data First?

A student asked me the question of whether or not scaling matters when running correlations. Let'southward investigate.

          library(tidyverse) library(knitr) set.seed(1)        

First, permit's become two random samples of integers equally our data:

          x = sample.int(20,20) y = sample.int(50,20) data.frame(x,y)%>%   kable()        

x	y
6	47
eight	11
11	32
xvi	6
iv	13
xiv	18
15	i
9	17
19	37
1	14
3	twenty
ii	24
xx	19
ten	7
5	thirty
7	39
12	28
17	4
18	35
thirteen	46

          data.frame(x,y)%>%   summary()        

                      ten               y          Min.   : ane.00   Min.   : 1.00    1st Qu.: five.75   1st Qu.:12.50    Median :ten.50   Median :19.fifty    Mean   :10.l   Mean   :22.forty    3rd Qu.:15.25   3rd Qu.:32.75    Max.   :20.00   Max.   :47.00                  

These data points for these \(x\) and \(y\) variables were generated from a uniform distribution, significant that any number is equally as probable to be picked as any other. Allow's plot these information points:

          df = data.frame(x,y) ggplot(df, aes(10,y))+   geom_point()+   geom_smooth(method = "lm")        

Every bit you lot tin can see, the data points are pretty much randomly scattered, only the ii variables are correlated past the slope of the line.

What happens if nosotros calibration the data?

          df$x_scaled = scale(df$x) df$y_scaled = scale(df$y) select(df, one_of(c("x_scaled", "y_scaled")))%>%   summary()        

                      x_scaled.V1          y_scaled.V1       Min.   :-1.6057931   Min.   :-ane.5438196    1st Qu.:-0.8028965   1st Qu.:-0.7141969    Median : 0.0000000   Median :-0.2092092    Hateful   : 0.0000000   Mean   : 0.0000000    third Qu.: 0.8028965   3rd Qu.: 0.7466604    Max.   : ane.6057931   Max.   : i.7746711                  

The data now has a mean of zero (and a standard deviation of 1; not shown). If we plot this data:

          ggplot(df, aes(x_scaled,y_scaled))+   geom_point()+   geom_smooth(method = "lm")        

We encounter no difference other than the fact that the axes accept moved to be centred at 0. We tin fifty-fifty compute the pearson correlation coefficient to show that both the unscaled and scaled information are not unlike in relation to each other:

          #role to compute correlation coefficient #unscaled cor(df$x,df$y)                  

          [ane] -0.05391063        

          #scaled cor(df$x_scaled, df$y_scaled)        

                      [,1] [one,] -0.05391063        

Same value.

---

But this has all been with a uniform sample; in the existent world, well-nigh of our information resembles a normal distribution. Let's have a sample of data from the normal distribution.

          a = rnorm(20,fifty, 10)%>%circular() b = rnorm(twenty, 30, 8)%>%circular() data.frame(a,b)%>%   summary()        

                      a               b          Min.   :30.00   Min.   :21.00    1st Qu.:46.00   1st Qu.:26.50    Median :49.00   Median :31.00    Hateful   :49.85   Mean   :31.10    3rd Qu.:56.fifty   third Qu.:35.25    Max.   :64.00   Max.   :46.00                  

          data.frame(a,b)%>%   kable()        

a	b
59	29
58	28
51	36
xxx	34
56	24
49	24
48	33
35	36
45	29
54	37
64	33
49	25
54	33
49	21
36	41
46	46
46	27
49	22
61	35
58	29

Now each drove of data \(a\) and \(b\) has a dissimilar hateful. The standard deviations are ten and five respectively likewise (non shown). Permit's plot these information:

          df2 = data.frame(a,b) ggplot(df2, aes(a,b))+   geom_point()+   geom_smooth(method = "lm")        

Yous can see the clear correlation here. At present let'southward scale these variables to see if that changes anything:

          df2$a_scaled = scale(df2$a) df2$b_scaled = scale(df2$b) select(df2, one_of(c("a_scaled", "b_scaled")))%>%   summary()        

                      a_scaled.V1          b_scaled.V1       Min.   :-2.2486702   Min.   :-one.5567091    1st Qu.:-0.4361401   1st Qu.:-0.7089962    Median :-0.0962907   Median :-0.0154130    Mean   : 0.0000000   Hateful   : 0.0000000    3rd Qu.: 0.7533329   3rd Qu.: 0.6396379    Max.   : 1.6029564   Max.   : 2.2965313                  

One time again, the mean has been brought downwards to 1 and the standard deviation is now besides 1 for both variables.

Now we plot:

          ggplot(df2, aes(a_scaled,b_scaled))+   geom_point()+   geom_smooth(method = "lm")        

Same story; the data looks exactly the same except for the fact that the axes are centred at 0.

Correlation coefficient confirms this:

          #unscaled cor(df2$a, df2$b)        

          [one] -0.2496821        

          #scaled cor(df2$a_scaled, df2$b_scaled)        

                      [,1] [i,] -0.2496821        

---

Remember that scaling is what's called a transformation. Some transformations practice wild things to information, just scaling is ane that does not alter the key qualities of the data (similar the shape of the spread), just it does modify the characteristics of information technology, like where the data really is in space. Scaling is done to standardise data, mostly so that when we read the data (and when we apply statistical methods to information technology), the two variables are more than comparable.

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