Standard deviation

Standard deviation shows how much data deviates from the mean.

Standard deviation

The standard deviation measures how far the values move from the mean on average. If it is small the data is tight; if large, spread out.

a = np.array([10, 10, 10, 10])
b = np.array([0, 5, 15, 20])
print(a.std(), b.std())   # 0.0 7.905...

Both arrays have the same mean (10) but very different spreads; the mean alone isn't enough.

ddof: sample and population

By default NumPy computes the population standard deviation (divides by n). When estimating the population from a sample, ddof=1 divides by n-1; that is the pandas default:

x = np.array([2, 4, 4, 4, 5, 5, 7, 9])
print(x.std())          # 2.0
print(x.std(ddof=1))    # 2.138...

With large datasets the difference becomes negligible; but when NumPy and pandas results differ, this is usually why.

Exercise Measure the spread

Print the population standard deviation (the default) of array x. Then print how many standard deviations each value is from the mean ((x - mean) / std).

Show example solution

import numpy as np

x = np.array([2, 4, 4, 4, 5, 5, 7, 9])

s = x.std()
print(s)
print((x - x.mean()) / s)

Test yourself

What is the standard deviation of an array whose values are all the same?
- 1
- Equal to the mean
- Undefined
- 0
Show answer

0 — No value deviates from the mean, so the spread is zero.
Compute the sample standard deviation.
```
s = x.std(____)
```
Show answer
```
s = x.std(ddof=1)
```
Match the values.

small std()

Data is tight
large std()

Data is spread out
ddof=0

Dividing by n
ddof=1

Dividing by n-1

Next lesson: Variance

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