Numerical Data Displays and Spread

Dot plots: every value on a number line

The bar graphs of the last lesson sorted data into categories, such as pets or seasons. Quiz scores are different. Each one is a number on a scale, so the picture should be built on a number line. A dot plot draws a number line and puts one dot above it for each value. When a value occurs more than once, its dots stack up in a column. (Some books call this a line plot. It is not the line graph of the last lesson.)

Here are the two classes, $11$ students each:

Each list adds up to $77$, and $77 \div 11 = 7$, so both means are $7$.

Reading a dot plot is counting. The number of dots is the number of values, $11$ in each class here. The tallest stack is the mode, so Class A’s mode is $7$, with five dots, and Class B’s mode is $9$, with three. (Stacks tied for tallest give several modes, and stacks all the same height give no mode, as in the lesson on mean, median, mode, and range.) Because every value stays visible, a dot plot also shows the shape of the data at a glance. Class A’s dots pile up in one tight group around $7$. Class B’s dots crowd together at the high end and trail off toward the low scores on the left.

To draw one by hand, mark a number line from the smallest value to the largest, with evenly spaced marks. Then place one dot per value, stacking repeats with equal gaps so that the stacks can be compared fairly.

Histograms: bars over number intervals

A dot plot gets crowded when there are many values and few of them repeat. Then it helps to group the values into intervals, equal stretches of the number line, and count how many values fall in each. A histogram draws one bar for each interval, and the height of the bar is the number of values in that interval.

A histogram looks like a bar graph, but its horizontal axis is different. A bar graph’s bars stand for categories, such as Dog and Cat, so they sit apart and could be put in any order. A histogram’s bars stand for intervals on a number line. Their order is fixed, from the smallest numbers to the largest, and the bars touch because each interval begins where the one before it ends.

Every whole number of minutes belongs to exactly one interval. That is why the labels read $0$ to $9$ and $10$ to $19$, and not $0$ to $10$ and $10$ to $20$: with those labels, a time of exactly $10$ minutes would belong to two bars. Every interval is also the same width. That is what lets the heights be compared fairly, and this course always uses equal widths.

Grouping has a cost. The histogram shows that $7$ students read for $20$ to $29$ minutes, but not which times they read. From the histogram alone you cannot find the longest time or the exact median, only the interval each one falls in.

Quartiles and the five-number summary

The median cuts sorted data into two halves. Cutting each half at its own middle splits the sorted list into four parts of nearly equal size. The three cut points are called the quartiles.

Start with Class B, already sorted. Its $11$ values have a single middle value, the $6$th, so the median is $8$. Leave that middle value out, and five values are left on each side of it. The lower half is the five values before it, $2, 4, 5, 6, 7$, and the upper half is the five values after it, $8, 9, 9, 9, 10$. The first quartile, $Q_1$, is the median of the lower half, and the third quartile, $Q_3$, is the median of the upper half:

So $Q_1 = 5$ and $Q_3 = 9$. The three cut points leave two values in each of four groups: $2$ and $4$, then $6$ and $7$, then $8$ and $9$, then $9$ and $10$. Each group holds about a quarter of the places in the sorted list, which is where the name quartile comes from. This works because each quartile is found exactly the way the median was. The median halves the whole list, and each quartile halves one of those halves.

With an even count, nothing is left out. The two middle values give the median, as in the lesson on the mean and median, and the list splits cleanly into a first half and a second half. Here is the whole method:

1. Sort the values from least to greatest.
2. Find the median.
3. Split the list into halves. With an even count, take the first half and the second half. With an odd count, leave the middle value out of both halves.
4. $Q_1$ is the median of the lower half, and $Q_3$ is the median of the upper half. A half with an even count takes the mean of its two middle values, just as the median does.

The list needs at least two values, or there are no halves to work with. Books and calculators do not all agree on step 3: some keep the middle value in both halves when the count is odd, which can give slightly different quartiles. Every quartile in this course leaves the middle value out.

The five-number summary lists the minimum, $Q_1$, the median, $Q_3$, and the maximum, in that order. For Class B it is $2, 5, 8, 9, 10$. The same steps give $5, 6, 7, 8, 9$ for Class A.

The interquartile range

The range measures spread using the two most extreme values. The interquartile range, or IQR, measures the spread of the middle of the data instead. Class A’s middle scores run from $Q_1 = 6$ to $Q_3 = 8$, a stretch $2$ points wide. Class B’s run from $5$ to $9$, a stretch $4$ points wide, twice as wide. That width is the IQR:

Like the range, the IQR measures spread, not center, and it is never negative, because $Q_3$ is never below $Q_1$. An IQR of $0$ is possible. For $1, 5, 5, 5, 5, 5, 5, 5, 12$, both quartiles are $5$, so the IQR is $0$, yet the range is $11$. The middle of that data is piled on one value, though the values outside the middle still differ.

The IQR has one big advantage over the range: a single extreme value hardly affects it.

A value that sits far from the rest of the data, like that $0$, is called an outlier. One outlier can stretch the range a long way while the IQR barely moves. So when a data set has an outlier, the IQR usually describes its typical spread better than the range does.

Box plots: five numbers as a picture

A box plot, also called a box-and-whisker plot, draws the five-number summary above a number line. To draw one:

1. Draw a number line that covers the minimum and the maximum, with evenly spaced marks.
2. Draw a box from $Q_1$ to $Q_3$, and a line across the box at the median.
3. Draw a whisker, a line from each side of the box out to the minimum and to the maximum.

In this lesson the whiskers always reach the minimum and the maximum. Some books stop a whisker early and mark far-off values as separate dots, using a rule for outliers that later courses teach.

The three lines of the box and the two whisker ends split the sorted list into four parts of about equal size, counted by place in the list. A longer section does not mean more values. Class B’s left whisker, from $2$ to $5$, is three times as long as its right whisker, from $9$ to $10$, yet each one reaches two of the eleven places besides its quartile: $2$ and $4$ on the left, $9$ and $10$ on the right. A long section means the values there are spread out, and a short one means they are packed together.

Repeated values change how many scores land in each section of the picture. When no value repeats, each section holds about a quarter of them and the box about half. But a value repeated at a quartile sits on that line, so a section can hold far more. Class A has $9$ of its $11$ scores from $Q_1 = 6$ to $Q_3 = 8$, because a second $6$ and a second $8$ sit on the box’s edges. A section can even have zero length, when repeated values make two of the five numbers equal; then every value in that section is the same number.

The box plot gives up detail for that summary. The dot plot shows that three students scored $9$, but the box plot shows neither that nor how many students there are.

Comparing data sets: center, spread, and shape

To compare two data sets, look at their centers, then their spreads, then their shapes. Box plots drawn on one shared number line make this quick.

Describing shape uses a few everyday words, and in this course they describe what you see rather than anything you calculate. A cluster is a group of values close together, like Class B’s six scores from $8$ to $10$, and a gap is a stretch with no values at all. When one side of the data stretches out much further than the other, the data is skewed toward that side. Class B has a long tail on the left, so it is skewed left. When the two sides look roughly like mirror images, as in Class A, the data is roughly symmetric. An outlier is judged by eye here: say which value looks far from the rest, and why. Later courses give a rule for exactly how far is far enough.

Common mistakes