Understanding Mean and Variance Changes

When a constant value is added to every data point in a set, it affects the central tendency (mean) but not the dispersion (variance).

Impact on Mean

Let the original data set be $X = \{x_1, x_2, ..., x_n\}$ with mean $\bar{x}$. If a constant $c$ is added to each value, the new data set is $Y = \{x_1+c, x_2+c, ..., x_n+c\}$.

The new mean $\bar{y}$ is calculated as:

$ \bar{y} = \frac{\sum_{i=1}^{n} (x_i + c)}{n} = \frac{\sum_{i=1}^{n} x_i + \sum_{i=1}^{n} c}{n} = \frac{\sum_{i=1}^{n} x_i}{n} + \frac{nc}{n} = \bar{x} + c $

Since the constant $c=20$ is non-zero, the mean changes (it increases by 20).

Impact on Variance

The variance measures the spread of data points around the mean. The variance ($\sigma^2$) is calculated using the squared differences from the mean.

For the new data set $Y$, the variance $\sigma_{new}^2$ is:

$ \sigma_{new}^2 = \frac{\sum_{i=1}^{n} ((x_i + c) - \bar{y})^2}{n} $

Substituting $\bar{y} = \bar{x} + c$:

$ \sigma_{new}^2 = \frac{\sum_{i=1}^{n} ((x_i + c) - (\bar{x} + c))^2}{n} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} $

This is identical to the original variance $\sigma^2$. Therefore, adding a constant does not change the variance.

Conclusion

Adding 20 to each value in the data set causes the mean to change, but the variance remains unchanged.