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This page sets out the method for histograms in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Histograms topic page.
When you are given a grouped frequency table with unequal widths and asked to draw a histogram.
When a histogram is given and you need frequencies (or the total, or an estimate for part of a class).
On a histogram, frequency is the area of each bar: Frequency = Frequency Density × Class Width.
Bars can have different widths. Always multiply the vertical height by the width of that specific bar.
Frequency Density = Frequency ÷ Class Width. This is plotted on the vertical axis.
To find the total frequency across several intervals, calculate each bar area and add them together.
A taller bar does not always represent more items. Compare bar areas, not bar heights.
The vertical axis represents frequency density, never raw frequency.
A bar chart compares separate categories, and the bars are usually the same thickness, so height can stand for frequency. A histogram is on a continuous scale. A class from 0 to 10 is twice as wide as a class from 10 to 15. If you draw both with height equal to the frequency, the wider class looks like more area than its count deserves, and the eye is misled.
Making area proportional to frequency fixes that. The wider class is drawn shorter, for the same frequency, so the areas match. That is why the vertical axis says frequency density and not frequency. Reading the height as if it were a bar-chart frequency is the error this topic is built to stop.
Method: 1. Check whether the class widths are equal. If they are not, height is not frequency. 2. Work out each class width as upper bound minus lower bound. 3. Use that width for that class only. 4. Remember the blocks touch. There are no gaps to skip.
You are given a frequency and a class, and you need the height. Divide. Width 6 and frequency 18 give density 18 ÷ 6 = 3. Width 8 and frequency 20 give density 20 ÷ 8 = 2.5. The division is exact. Do not round a value that lands on a half or a simple decimal, and do not round an integer at all.
The width is the difference of the bounds, not the larger bound. A class from 6 to 10 has width 4, not 10 and not 6. Using the endpoint as the width makes the density far too small and the block far too short. Write the subtraction down even when it looks obvious.
Method: 1. Class width = upper bound − lower bound. 2. Frequency density = frequency ÷ class width. 3. Plot or read that density as the height of the block. 4. Keep exact values. 18 divided by 6 is exactly 3, so there is nothing to round.
The reverse question gives the height and the width and hides the frequency. Multiply. Density 4.5 and width 8 give frequency 4.5 × 8 = 36. Density 7 and width 6 give 7 × 6 = 42. You are finding the area. A calculator is appropriate, and the numbers in these questions are chosen so the product is exact.
Do this once per class. A common slip is to multiply every density by the same width, usually the first width you saw. Each block has its own width. A density of 3 on a width of 4 is frequency 12. The same density on a width of 8 is frequency 24. Same height, double the area, double the frequency.
Method: 1. Read the frequency density of the class. That is the height. 2. Find the width of that same class. 3. Frequency = density × width. 4. Repeat with the next class. Do not reuse a width.
Because area is proportional to frequency, you can compare blocks without a scale in your head. The block with twice the area has twice the frequency, even if it is lower and wider. Adding frequencies is adding areas, not adding heights. A total frequency is the sum of (density × width) over every class you were asked to include.
A missing frequency is the same multiply, once you know that class width. A missing density is the divide. After you have the frequencies, a total is ordinary addition. None of those steps treats the height as the count. If the widths had all been equal you would still be safer doing density × width, because the method does not change when a later class breaks the pattern.
Method: 1. For each class, multiply density by that class width. 2. Add those frequencies for a total. 3. Equal areas mean equal frequencies, whatever the heights. 4. Never copy the height into the frequency column when the widths differ.
When a question says "Draw a histogram" it gives a grouped frequency table with classes of different widths. Add two columns: class width and frequency density. Work out each width as upper boundary minus lower boundary, then divide the frequency by the width. Classes 0 < t ≤ 10 and 10 < t ≤ 30 have widths 10 and 20.
The bars are drawn with these densities as heights. In the exam you are often asked only for the densities or for one bar height. Check each one by multiplying back: density times width must return the frequency. If the question gives a total, add the frequencies and check they match. Label the vertical axis "Frequency density", not "Frequency".
Method: 1. Write each class width: upper boundary minus lower boundary. 2. Divide each frequency by its width to get the frequency density. 3. Multiply each density by its width to check you get the frequency back. 4. Add the frequencies and check against the total, if one is given.
Tip: Each width comes from that class only. Do not reuse the width of the class next to it.
A histogram assumes the data are spread evenly across each class. That lets you estimate the frequency for part of a class. For the class 30 < s ≤ 50 with density 3.6, the part 40 < s ≤ 50 has width 10, so the frequency is 3.6 × 10 = 36. Use only the width of the part you want, not the width of the whole class and not the boundary value itself.
Questions such as "Estimate the number of cars over 40 mph" combine this with whole bars. Work out the part-bar, then add the frequencies of all the whole bars above it. Show every product, as each line earns a mark. Write "estimate" in your answer, because the true count depends on how values are spread inside the class.
Method: 1. Mark the value you are interested in on the horizontal axis and see which class it falls in. 2. Work out the width of the part of that class you need. 3. Multiply that width by the frequency density of the class. 4. Add the whole bars beyond it, and write the total as an estimate.
Some questions leave the frequency density scale off, or draw a bar without numbers. The key idea is that frequency is proportional to area. If you know the frequency of one bar, divide it by its area to get the frequency per unit of area. Then multiply by the area of any other bar. Area is width times height, with the height measured on the drawing.
Another version gives the total frequency and asks for the height of a missing bar. Work out the frequencies of the known bars, subtract from the total, and divide the missing frequency by that class width. The result is the height. State the units or the axis label so the examiner can see what your number means.
Method: 1. Work out the area of the known bar as width times height. 2. Divide its frequency by its area to find the frequency per unit of area. 3. Work out the area of the other bar and multiply by this scale. 4. For a missing bar, subtract known frequencies from the total, then divide by the class width.
Tip: The frequency of a bar is its area, so a bar twice as wide and half as tall holds the same number of results.
To estimate the median, turn the histogram into frequencies, find the total, and halve it. Work along the cumulative total to find the class holding that position. The median is the lower boundary plus the fraction of the way through the class multiplied by its width. For a position 10 places into a class of 50 and width 20, add 10/50 × 20 to the lower boundary.
To estimate the mean, use the midpoint of each class. Multiply each midpoint by its frequency, add these products, and divide by the total frequency. Using midpoints is an assumption, so the answer is an estimate. Set the working out in a table so each line is clear, and give the answer to the accuracy asked, such as one decimal place.
Method: 1. Find every class frequency by multiplying density by width. 2. For the median, halve the total and locate the class using running totals. 3. Interpolate inside the class by proportion. 4. For the mean, add up (midpoint times frequency) and divide by the total frequency.
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