5.2 Measures of center & spread

Summarizing a Dataset with a Single Number

A graph shows shape; a single number summarizes center or spread. The TSIA2 tests three measures of center (mean, median, mode) and three common measures of spread (range, IQR, standard deviation).

Measures of Center

• Mean (average) — add all the values, then divide by how many there are.
• Median — the middle value once the data are sorted; with an even count, average the two middle values.
• Mode — the value that appears most often (a dataset may have no mode or several).

Worked example: Seven quiz scores are 8, 6, 9, 10, 6, 7, 12.

Mean: sum = 8 + 6 + 9 + 10 + 6 + 7 + 12 = 58. Divide by 7: 58 / 7 ≈ 8.29.

Median: sort the values → 6, 6, 7, 8, 9, 10, 12. With seven values the 4th is the middle, so the median = 8.

Mode: 6 appears twice and every other value once, so the mode = 6.

Even-count median: for the six values 6, 6, 7, 8, 9, 10 the middle two are 7 and 8, so the median = (7 + 8) / 2 = 7.5.

Measures of Spread

Range = maximum - minimum. For the seven scores, 12 - 6 = 6. Range uses only two values, so a single extreme changes it a lot.

Quartiles and IQR — Q1 is the median of the lower half and Q3 is the median of the upper half. The interquartile range IQR = Q3 - Q1 covers the middle 50% and ignores the extremes. Using the sorted list 6, 6, 7, 8, 9, 10, 12 (median 8 is the 4th value): the lower half 6, 6, 7 gives Q1 = 6; the upper half 9, 10, 12 gives Q3 = 10. So IQR = 10 - 6 = 4.

Standard deviation (conceptual) — the typical distance of a value from the mean. A small standard deviation means values cluster near the mean; a large one means they are spread out. You rarely compute it by hand on the TSIA2, but you should know that a standard deviation of 0 means every value is identical, and adding an outlier makes it larger.

Building a Five-Number Summary

Put the data in order, find the median, then find Q1 and Q3 as the medians of the lower and upper halves. For the ten values 3, 5, 7, 8, 10, 12, 15, 18, 20, 25, the median is the average of the 5th and 6th values: (10 + 12) / 2 = 11. The lower half 3, 5, 7, 8, 10 has median Q1 = 7, and the upper half 12, 15, 18, 20, 25 has median Q3 = 18. So the five-number summary is 3, 7, 11, 18, 25, the range is 25 - 3 = 22, and the IQR is 18 - 7 = 11. These five numbers are exactly what a box plot draws.

The Effect of Outliers

An outlier is a value far from the rest. Because the mean uses every value, one large outlier drags it toward the extreme; the median barely moves because it depends only on position.

Worked example: Five incomes (in thousands) are 40, 42, 45, 48, 50. The mean = (40 + 42 + 45 + 48 + 50) / 5 = 225 / 5 = 45, and the median = 45 (the 3rd value). They agree.

Now replace 50 with an outlier of 200: the values become 40, 42, 45, 48, 200. The mean = (40 + 42 + 45 + 48 + 200) / 5 = 375 / 5 = 75, but the median = 45 (still the 3rd value). The mean jumped from 45 to 75 while the median stayed at 45. This is why the median is called resistant, and why home prices and salaries are usually reported with the median.

Comparing Distributions

To compare two groups, look at both center and spread:

• A higher center (mean or median) means the values are typically larger.
• A larger spread (range, IQR, or standard deviation) means the group is less consistent.

Skew hint: when data are skewed right (a high tail), the mean is pulled above the median; skewed left pulls the mean below the median; in a symmetric distribution mean ≈ median.

Worked example: Class A scores average 80 with an IQR of 6, and Class B scores average 80 with an IQR of 20. The centers match, but Class A is far more consistent — its middle 50% spans only 6 points versus 20 for Class B.

Choosing the Right Measure

• Roughly symmetric with no outliers → use the mean and standard deviation.
• Skewed or containing outliers → use the median and IQR.
• Categorical data → use the mode, the only center that applies.