Histogram Heights

Histogram Heights

A histogram looks like a bar chart and obeys a different rule, and nearly every mark on this topic turns on knowing which rule you are under. On a bar chart the height tells you the frequency. On a histogram the area does. That difference exists because histograms are used when the class intervals are different widths, which happens whenever data has a long thin tail and splitting it into equal classes would leave a row of nearly empty bars. Once widths differ, comparing heights compares nothing, and something else has to carry the frequency. (Higher tier.)

Collecting the data

Before any of it can be represented it has to be gathered, and how you gather decides what the graph is worth:

Which criticism is strongest?

A gym wants to know how satisfied its members are. It hands a questionnaire to everyone using the gym between 6 and 7 on a Tuesday evening and gets 80 responses. Which is the strongest criticism?

• It only reaches people who came that evening, so the most frequent attenders are the most likely to be included and the members who rarely come, or who stopped coming because they were dissatisfied, are missed entirely
• The sample of 80 is too small to draw any conclusion from
• It was taken in the evening, so people who prefer mornings were missed
• A questionnaire is not a reliable way to measure satisfaction

Two charts, two rules

They look alike on the page and they are read completely differently:

Find the frequency density

A class covers 40 ≤ t < 70 and contains 30 values. What is its frequency density?

Work backwards

On a histogram, a bar has a frequency density of 2.5 and covers a class of width 20. How many values does it represent?

A histogram to read

Times taken to complete a task, in minutes. Only the class boundaries and the frequencies are given, because on a real exam paper that is all you get: • 0 ≤ t < 10, frequency 15 • 10 ≤ t < 20, frequency 30 • 20 ≤ t < 40, frequency 40 • 40 ≤ t < 70, frequency 30 One worked, as the model. The first class runs from 0 to 10, so its width is 10. Its frequency density is 15 ÷ 10 = 1.5, and that is the height its bar would be drawn at. Work the other three out for yourself before going on. Note that the widths are 10, 10, 20 and 30, so this is genuinely a histogram and not a bar chart in disguise, and 115 people took part in total.

Match each class to its frequency density

Where are the most people?

This histogram shows the data above. Tap the bar for the class containing the most people.

Part of a class

Using the same histogram, estimate how many people took between 25 and 40 minutes.

What examiners are watching for

Label the axis "frequency density". Writing "frequency" there is a lost mark on a question you otherwise answered correctly. Leave no gaps. Continuous classes run into one another, so the bars must touch. Part of a class is part of the area. To estimate a range inside a class, take the frequency density and multiply by the width of the part you want, not the width of the whole class. Between 25 and 40 in a class of density 2 is 2 × 15, not 2 × 20. And know what you assumed. That calculation only works if the values are spread evenly across the class, and there is no guarantee they are. If most of the 20-to-40 group actually finished near 22 minutes, the true answer is lower. This is why such answers are called estimates, and saying so is worth a mark on an "explain why this is only an estimate" question.

Why only an estimate?

You calculated that 30 people took between 25 and 40 minutes. The question then asks why this can only be an estimate. What is the reason?

• The calculation assumes the 40 people in the 20-to-40 class are spread evenly across it. The histogram does not record where inside the class each value actually fell, so if they clustered near one end the true figure would be different
• Because reading the frequency density off the axis by eye is never completely accurate
• Because frequency densities are usually rounded, so the answer carries that rounding forward
• Because the data came from a sample rather than the whole population

Draw a histogram

You are given a grouped frequency table with unequal classes. Put the steps for drawing the histogram into order.

• Work out the width of each class from its boundaries
• Divide each frequency by its class width to get the frequency density
• Choose a vertical scale that fits the largest density, and label the axis "frequency density"
• Draw each bar across the full width of its class, at the height of its density
• Check that the bars touch, with no gaps between them

The histogram rules

Histograms are used when the classes have unequal _____, which is exactly when comparing heights would mislead. The vertical axis shows frequency density, worked out as frequency divided by class width, and the frequency of a class is shown by the _____ of its bar. So the tallest bar is not necessarily the biggest class. To find part of a class you multiply the density by the width of the _____ you want, and the answer is only an estimate because it assumes the values are spread _____ across the class.