Measures of Center and Spread
Calculate mean, median, mode, range, interquartile range, and standard deviation. Understand the effect of outliers and skew on each measure and choose the appropriate summary statistic for a given context.
A lesson of Algebra 1, in Data Analysis & Statistics, about 35 minutes.
By the end of this lesson, you will be able to compute mean, median, mode, range, IQR and standard deviation, say which measure an outlier wrecks, and choose the right summary for a given set of data.
Center: Three Different Questions
"Typical" is not one idea, and the three measures answer three different questions.
Mean. Add everything, divide by how many. The balance point.
$$\bar{x} = \frac{\sum x}{n}$$
Median. Sort, then take the middle value. With an even count, average the two middle ones. The halfway point by position.
Mode. The most frequent value. There can be none, one, or several.
$$2, 3, 3, 4, 8$$
Mean $= 4$, median $= 3$, mode $= 3$. Three answers to "what is typical", all correct, describing different things.
Sort before finding a median, every time. The middle of an unsorted list is meaningless, and forgetting is the commonest arithmetic slip in the topic.
Outliers Move the Mean and Not the Median
$$2, 3, 3, 4, 8 \quad \Rightarrow \quad \text{mean } 4, \text{ median } 3$$
$$2, 3, 3, 4, 88 \quad \Rightarrow \quad \text{mean } 20, \text{ median } 3$$
One changed value dragged the mean from $4$ to $20$. The median did not move at all.
The mean uses every value's SIZE; the median uses only their ORDER. So an extreme value has unlimited influence on one and none on the other, and that is the single most useful fact in this lesson.
Which to report:
- Symmetric data, no outliers: the mean, because it uses all the information
- Skewed data or outliers present: the median, because it is resistant
House prices and incomes are always reported as medians, and this is why. A few very large values pull the mean above what almost anyone actually has, so the mean is a true number that describes nobody.
Sort It: Which Measure Should Be Reported
Example 1: All Three Measures
Here is every step, worked all the way through. $$7, 3, 9, 3, 8, 12$$
- Sort them and write the largest value 12
- Add all six values. What is the total? 42
- Divide by 6. What is the mean? 7
- There are six values, so average the two middle ones, 7 and 8. What is the median? 7.5
- Which value appears twice? That is the mode 3
Spread: How Scattered Is It
Two data sets can share a mean and be nothing alike.
$${50, 50, 50} \qquad \text{and} \qquad {0, 50, 100}$$
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