• The average alone does not always provide a complete understanding of a dataset.
• Variability measures how much the values in a dataset differ from one another.
• Two datasets can have the same average but different amounts of variation.
• Variance and standard deviation help us understand how spread out the values are.

• Variance measures how spread out the values are from the mean.
• A higher variance means the values are more spread out.
• A lower variance means the values are closer to the mean.

Sample Variance Formula

\[ s^2 = \frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1} \] \(x_i\) = Individual value

\(\bar{x}\) = Sample mean

\(n\) = Number of observations

• Standard deviation measures how much the values in a dataset typically differ from the mean.
• It is calculated by taking the square root of the variance.
• A lower standard deviation indicates that the values are closer to the mean.
• Standard deviation uses the same units as the original data.

Sample Standard Deviation Formula

\[ s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}} \]

Consider two groups of people with the following hours of sleep:

Both groups have an average of 7 hours of sleep. However, Group B has greater variation in its sleep hours, which results in a higher standard deviation.

The graph below compares the hours of sleep between both groups.

Group B has a greater variation in hours of sleep compared to Group A.

Both groups have an average of 7 hours of sleep. However, Group B’s values are more spread out from the mean, resulting in a higher standard deviation compared to Group A.

The graph shows that Group B has greater variation in hours of sleep compared to Group A.

We can use R to calculate the variance and standard deviation for both groups.

var(group_a)
[1] 1
var(group_b)
[1] 9

sd(group_a)
[1] 1
sd(group_b)
[1] 3

Group B has a higher variance and standard deviation, showing that its hours of sleep are more spread out.

• Variance and standard deviation help us understand how much the values in a dataset differ from one another.
• Two datasets can have the same average but different amounts of variation.
• In the example, Group B had a higher variance and standard deviation, showing that its hours of sleep were more spread out.
• Understanding variance and standard deviation allows us to better interpret data rather than relying only on the average.