TheIBAxolotl
Measures of centre
- Mean \(\bar{x} = \dfrac{\sum f x}{\sum f}\): uses every value, but is pulled by outliers.
- Median: the middle value of the ordered data; resistant to outliers, so it suits skewed data such as incomes.
- Mode: the most frequent value; the only one that works for categories.
Grouped data
When data are grouped, the exact values are lost, so the mean is an estimate. Replace each class by its mid-interval value and use the frequencies. The modal class is the class with the highest frequency (for equal widths).
Measures of spread
- Range = max − min: simple but depends only on the extremes.
- IQR = \(Q_3 - Q_1\): the spread of the middle half; resistant to outliers.
- Standard deviation \(\sigma\): roughly the typical distance of values from the mean. Its square is the variance:
\[\sigma^2 = \frac{\sum f (x - \bar{x})^2}{\sum f}\]
In this course the standard deviation is found with technology, and it is the population version (\(\sigma_x\) on most calculators, not \(s_x\)).
Worked example
Waiting times (minutes):
| Time \(t\) | \(0 \le t < 10\) | \(10 \le t < 20\) | \(20 \le t < 30\) |
|---|---|---|---|
| Frequency | 4 | 10 | 6 |
Midpoints 5, 15, 25. Estimated mean \(= \dfrac{4(5) + 10(15) + 6(25)}{20} = \dfrac{320}{20} = 16\) minutes. Entering midpoints and frequencies in the GDC gives \(\sigma = 7\) minutes (check: mean of squares \(= 305\), so \(\sigma^2 = 305 - 16^2 = 49\)). The modal class is \(10 \le t < 20\).
Effect of constant changes
If every value is changed in the same way:
| Change | Mean, median, mode | SD, IQR, range | Variance |
|---|---|---|---|
| add \(c\) | \(+ c\) | unchanged | unchanged |
| multiply by \(k\) (\(k>0\)) | \(\times k\) | \(\times k\) | \(\times k^2\) |
For example, if every worker's wage rises by USD 20 a day, the mean wage rises by USD 20 and the standard deviation stays the same; a 5% pay rise multiplies both by 1.05.
Finding a missing value
If the mean of a set is known, use \(\sum x = n\bar{x}\) to find an unknown value or frequency. For example, if five numbers have mean 12 and four of them add to 47, the fifth is \(60 - 47 = 13\).
GDC tips
- Put values in one list and frequencies in another, then run one-variable statistics with both lists.
- Read \(\bar{x}\), \(\sigma_x\), \(n\), min, \(Q_1\), median, \(Q_3\), max from the same screen.
Common mistakes
- Using class boundaries instead of midpoints for a grouped mean.
- Reporting \(s_x\) instead of \(\sigma_x\).
- Forgetting the frequency list, so the GDC treats each midpoint once.
- Saying adding a constant changes the standard deviation.
Exam tips
- Write "estimate" for grouped means and say why (exact values unknown).
- Choose the median and IQR when data are skewed or contain outliers, and say so.
- Variance questions usually want \(\sigma^2\): square the GDC value.