Measures of Centre, Spread and Data Transformations
Measures of Centre, Spread and Data Transformations
| Measure | Interpretation or calculation |
|---|---|
| Mean | \(\overline{x}=\dfrac{\sum\limits_i f_i x_i}{\sum\limits_i f_i}\); affected by extreme observations. |
| Median | Middle of the ordered data; useful when extremes distort the mean. |
| Mode | Most frequent value; there may be several modes or none. |
| Variance and standard deviation | Use one-variable statistics with frequencies. Variance is the square of standard deviation. Unless stated otherwise, treat the data as the population. |
| Quartiles | Use the required calculator convention. Hand and calculator conventions can give different quartiles for a discrete data set. |
Example: for \(2,4,4,6\), the mean and median are \(4\), the mode is \(4\), the population variance is \(2\), and the population standard deviation is \(\sqrt2\).
| Every value becomes | Mean becomes | Standard deviation becomes | Variance becomes |
|---|---|---|---|
| \(x+b\) | \(\overline{x}+b\) | \(\sigma\) | \(\sigma^2\) |
| \(ax+b\) | \(a\overline{x}+b\) | \(|a|\sigma\) | \(a^2\sigma^2\) |
Changing every value is different from correcting one observation. For a negative scale factor the order reverses, although range and IQR still scale by its absolute value.
Choose the summary for the question being asked. The mean uses every observation, while the median depends on the ordered middle. Compare centre and spread separately and include units: a lower mean waiting time and a smaller standard deviation describe shorter waits and greater consistency respectively.
For frequencies, the total number of observations is \(n=\sum f\), not the number of distinct values. Enter the frequency list explicitly. For grouped intervals use their midpoints; both the mean and standard deviation obtained from those midpoints are estimates.
Keep full precision until the final answer. A correction to one observation changes the totals used for the mean and variance; the affine-transformation rules apply only when every observation undergoes the same change.
For values \(x_i\) with frequencies \(f_i\), let \(n=\sum_i f_i\). The population standard deviation is \(\sigma=\sqrt{\dfrac{\sum_i f_i x_i^2}{n}-\overline{x}^{\,2}}\). For ungrouped observations each frequency is one. For grouped intervals use \(m_i=\dfrac{\text{lower boundary}+\text{upper boundary}}{2}\) in place of \(x_i\); the resulting mean and standard deviation are estimates. Grouping does not reveal the exact smallest and largest observations.
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