9.1 Measures of Central Tendency & Measures of Variation
Key Takeaways
- The mean is the arithmetic average, the median is the middle value of ordered data, and the mode is the most frequent value; a data set may have no mode or several.
- The median resists outliers while the mean does not, which is why median income is reported instead of mean income.
- Range is the simplest measure of variation but depends only on the two extreme values; the interquartile range Q3 − Q1 describes the middle 50% and resists outliers.
- Mean absolute deviation averages the distances of the data values from the mean, giving a spread measure in the original units.
- Adding a constant to every value shifts the mean and median by that constant but leaves range, IQR, and MAD unchanged.
9.1 Measures of Central Tendency & Measures of Variation
Competency 4 rose from 7% to 15% of the test when Florida redeveloped test 025, so data analysis now carries as much weight as number sense. Skill 1 asks for both center and spread — reporting a mean without a spread measure is an incomplete description of a data set, and items test that idea directly.
Measures of center
| Measure | Definition | Best when |
|---|---|---|
| Mean | Sum of values ÷ number of values | Data are roughly symmetric with no outliers |
| Median | Middle value of ordered data | Data are skewed or contain outliers |
| Mode | Most frequently occurring value | Data are categorical, or you want the most common response |
Computing the median requires ordering the data first — the most-skipped step. With an odd count, the median is the middle value. With an even count, it is the mean of the two middle values.
Data: 12, 7, 19, 4, 15, 9. Ordered: 4, 7, 9, 12, 15, 19. Six values, so the median is (9 + 12)/2 = 10.5.
A set may have no mode (all values distinct), one mode (unimodal), or two or more (bimodal, multimodal). The mode is the only measure of center that works for categorical data such as favorite color.
Choosing the right measure
[!IMPORTANT] The mean is pulled toward outliers; the median is not. This single property drives most item design in this skill.
Salaries: $38k, $41k, $43k, $45k, $290k. Mean = 457/5 = $91.4k — higher than four of the five salaries. Median = $43k — a far better description of a typical employee.
This is why median household income is the standard public statistic. A skewed distribution has its mean pulled toward the long tail: in a right-skewed set, mean > median; in a left-skewed set, mean < median; in a symmetric set, mean ≈ median. Items give a skew description and ask which measure is larger, or ask which measure best represents the data.
Weighted mean. When values represent groups of different sizes, weight them.
A class of 20 averaged 78 and a class of 30 averaged 88. Combined mean = (20 · 78 + 30 · 88)/50 = (1,560 + 2,640)/50 = 4,200/50 = 84. Averaging 78 and 88 to get 83 ignores that the second class is larger — the standard distractor.
Working backward from a mean is a standard item type.
Six test scores average 82. After a seventh test the average is 84. What was the seventh score? Old total = 6 × 82 = 492. New total = 7 × 84 = 588. The seventh score = 588 − 492 = 96.
Measures of variation
Range = maximum − minimum. Simple, but it uses only two values and is destroyed by a single outlier.
Quartiles and the interquartile range. Order the data, find the median (Q2), then find the median of the lower half (Q1) and of the upper half (Q3).
IQR = Q3 − Q1, the spread of the middle 50% of the data.
Data: 3, 7, 8, 12, 14, 18, 21, 25, 30 Median (Q2) = 14. Lower half: 3, 7, 8, 12 → Q1 = (7 + 8)/2 = 7.5. Upper half: 18, 21, 25, 30 → Q3 = (21 + 25)/2 = 23. IQR = 23 − 7.5 = 15.5
Because the IQR ignores the extreme quarters, it is resistant to outliers, making it the spread measure that pairs naturally with the median.
Outlier rule. A value is commonly flagged as an outlier if it falls below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).
With Q1 = 7.5, Q3 = 23, IQR = 15.5: the fences are 7.5 − 23.25 = −15.75 and 23 + 23.25 = 46.25. No value in the set above lies outside those bounds, so there are no outliers.
Mean absolute deviation (MAD). Average how far each value sits from the mean.
Data: 4, 8, 10, 14. Mean = 36/4 = 9. Distances from 9: 5, 1, 1, 5. Sum = 12. MAD = 12/4 = 3.
MAD is expressed in the original units, so "MAD = 3 points" means values sit an average of 3 points from the mean. Use absolute distances — signed deviations always sum to zero, which is why the absolute value is required.
A larger MAD means more spread. Two classes with the same mean but MADs of 2 and 9 differ sharply in consistency, and comparison items lean on exactly that reading.
Effect of transformations
+---------------------------------------------------------------------------+
| ADD c to every value: |
| mean +c median +c mode +c |
| range UNCHANGED IQR UNCHANGED MAD UNCHANGED |
| |
| MULTIPLY every value by c: |
| mean xc median xc mode xc |
| range xc IQR xc MAD x|c| |
+---------------------------------------------------------------------------+
The reasoning: adding a constant slides the whole distribution without changing any distance between values, so every spread measure is untouched. Multiplying stretches all distances proportionally, so every spread measure scales.
Every score on a test is raised by 5 points. The mean rises by 5, and the standard spread measures are unchanged. Every measurement in centimeters is converted to millimeters by multiplying by 10. Both the mean and the range multiply by 10.
A data set is 6, 9, 11, 14, 15, 19, 22, 28. What is the interquartile range?
Five homes on a street sold for $215,000, $228,000, $241,000, $236,000, and $1,450,000. Which measure best represents a typical sale price, and why?
Every student in a class receives 4 bonus points added to their test score. What happens to the mean and to the mean absolute deviation?