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Histograms - Edexcel GCSE Higher Maths

Typically 3-6 marks in one structured question when it appears · Spec 1MA1

A histogram looks like a bar chart until you check the vertical axis. On a histogram the area of a bar is the frequency, so the height has to be the frequency shared out over the width of the class, and that height is called the frequency density: frequency density=frequencyclass width\text{frequency density} = \frac{\text{frequency}}{\text{class width}}. The adjustment is what allows classes of different widths onto the same diagram without the wide ones dominating it. On Edexcel 1MA1 this is spec ref S3 and it is Higher tier only, so a Foundation candidate draws bar charts and frequency polygons and never meets frequency density.

Edexcel sets it as a single question of two or three parts, and the parts run in opposite directions. One gives you a grouped frequency table and asks for the histogram, which means building a frequency density column before you touch the grid. The other gives you the histogram and asks you to complete the table, or to work out how many items fall in a range that cuts through the middle of a bar. It can appear on any of the three papers, and the calculator changes very little, because the arithmetic is divisions like 44 by 20 and multiplications like 15 by 2.2.

The marks go in predictable places. Mark schemes hold something for any clear use of frequency density, so writing the divisions in a column beside the table earns credit even when a bar ends up at the wrong height, and the same column is what makes the vertical axis label obvious. Two errors turn up on this question every series: some candidates draw a bar chart with the raw frequencies as heights and score nothing, and another group reads a bar height straight off the axis and writes it down as a frequency. Multiply the height by the class width before you record any count, and check that your frequencies add back to the total the question gave you.

Worked example

The table gives information about the times, in minutes, that 110 customers spent in a shop. Time 0<t≤100 < t \le 10: 15 customers. Time 10<t≤2010 < t \le 20: 30 customers. Time 20<t≤4020 < t \le 40: 44 customers. Time 40<t≤7040 < t \le 70: 21 customers. (a) Work out the frequency density for each class. (b) Estimate the number of customers who spent between 25 and 40 minutes in the shop. (c) Estimate the number of customers who spent more than 25 minutes in the shop.
[5 marks]
  1. Write the class widths down first: 10−0=1010 - 0 = 10, 20−10=1020 - 10 = 10, 40−20=2040 - 20 = 20, 70−40=3070 - 40 = 30. They are not all the same, which is why frequency density is needed at all.
    M1: class widths from the boundaries, not from the midpoints
  2. Divide each frequency by its own class width: 15÷10=1.515 \div 10 = 1.5, 30÷10=330 \div 10 = 3, 44÷20=2.244 \div 20 = 2.2, 21÷30=0.721 \div 30 = 0.7.
    A1: all four frequency densities 1.5, 3, 2.2, 0.7; these are the bar heights
  3. For (b), 25 to 40 minutes sits inside the class 20<t≤4020 < t \le 40, whose bar has height 2.2. The part wanted is 15 minutes wide.
    M1: the correct frequency density and the correct width of the part, 40−25=1540 - 25 = 15
  4. Frequency == frequency density ×\times width =2.2×15=33= 2.2 \times 15 = 33 customers.
    A1: 33; an estimate, because the times inside a class are assumed evenly spread
  5. For (c), add the whole of the last class to that part: 33+21=5433 + 21 = 54 customers spent more than 25 minutes.
    B1: 54, following through from part (b)

Practice questions

These are original questions in Edexcel 1MA1 Higher style. There are no diagrams here, so each histogram is given as a list of its bars: either the class with its frequency density, or the class with its frequency so that you work the density out. Questions whose answer depends on splitting a class say so and ask for an estimate, exactly as the mark scheme treats them.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [1 mark]
    On a histogram, the bar for the class 10≤x<1510 \le x < 15 has a frequency density of 6. Work out the frequency for this class.
  2. Question 2 Warm-up [1 mark]
    Which of these is the correct rule for the height of a bar on a histogram?
    Answer options for question 2
  3. Question 3 Exam pace [2 marks]
    On a histogram the bar for the class 30<m≤5030 < m \le 50 represents 24 items. The bar for the class 50<m≤5550 < m \le 55 is drawn at exactly the same height. How many items are in the class 50<m≤5550 < m \le 55?
  4. Question 4 Exam pace [3 marks]
    A histogram is drawn for 200 items. The bar over 0<x≤40 < x \le 4 has a frequency density of 12.5, the bar over 4<x≤104 < x \le 10 has a frequency density of 9, and the bar over 10<x≤3010 < x \le 30 has a frequency density of hh. There are no other bars. Work out the value of hh.
  5. Question 5 Stretch [4 marks]
    A histogram is drawn for the weights of all the melons in a field, but the vertical axis is not labelled. The bar over 8<w≤98 < w \le 9 kg has a height of 2 units, the bar over 9<w≤119 < w \le 11 kg has a height of 3.5 units, and the bar over 11<w≤1411 < w \le 14 kg has a height of 1 unit. There are 24 melons weighing more than 8 kg and at most 9 kg. Work out the total number of melons in the field.
  6. Question 6 Stretch [4 marks]
    A histogram of the times, in minutes, taken by 120 people to complete a puzzle has these bars: 0<t≤100 < t \le 10, frequency density 1.8; 10<t≤2010 < t \le 20, frequency density 3; 20<t≤4020 < t \le 40, frequency density 2.4; 40<t≤6040 < t \le 60, frequency density 1.2. Work out an estimate for the mean time, in minutes.

Common mistakes examiners see

  • Drawing the frequencies as the bar heights, so a class of 44 items over a width of 20 is drawn at 44 instead of 2.2 and the diagram is a bar chart with unequal widths.

    Add a frequency density column to the table before you draw anything. If every class width is not the same number, the heights cannot be the frequencies, and the axis has to be labelled Frequency density.

  • Dividing every frequency by the same number, usually the width of the first class, so classes of width 10, 10, 20 and 30 all get divided by 10.

    Work out each class width separately and write it next to its row. The widths in an Edexcel table are deliberately mixed, and using one divisor throughout is the error examiner reports name most often.

  • Reading a bar height off the vertical axis and recording it as the frequency, so a bar of height 2.4 spanning a class of width 15 is written into the table as 2.4 rather than 36.

    On a histogram the area is the frequency. Multiply back: frequency == frequency density ×\times class width. A frequency in a table of people or parcels should come out as a whole number, which is a quick check that you multiplied rather than copied.

  • Dividing the wrong way round, so a class of width 15 containing 6 items gives 15÷6=2.515 \div 6 = 2.5 instead of 6÷15=0.46 \div 15 = 0.4.

    Frequency density is frequency per unit, so the frequency goes on top. Writing it as the fraction 615\frac{6}{15} rather than as a keyed division makes the order hard to get wrong, and it also survives the non-calculator paper.

  • Taking the whole class frequency when only part of the class is asked for, answering 44 for 'between 25 and 40 minutes' when the class runs from 20 to 40, or halving the class because it looks like about half.

    Find the frequency density of that class, then multiply by the width of the part you actually want: 2.2×15=332.2 \times 15 = 33. Say in your answer that it is an estimate, since the method assumes the values are spread evenly across the class.

Frequently asked questions

Are histograms on the Foundation paper?
No. Histograms with unequal class widths are Higher tier only on Edexcel 1MA1, listed under spec ref S3 alongside cumulative frequency and box plots. Foundation candidates work with bar charts, pictograms and frequency polygons, where the height of a bar is the frequency.
Is the frequency density formula given in the exam?
No. The Exam Aid formulae sheet issued with each 1MA1 paper covers areas and volumes, the quadratic formula, trigonometry and the probability rules, and it does not include frequency density. You have to recall that frequency density is the frequency divided by the class width, and that frequency is the frequency density multiplied by the class width.
What is the difference between a histogram and a bar chart?
A bar chart shows categories, leaves gaps between the bars and puts frequency up the vertical axis. A histogram shows continuous data on a numerical scale, the bars touch, the widths can differ, and the vertical axis is frequency density so that the area of each bar gives the frequency.
Why is the answer from a histogram only an estimate?
Because the original values were lost when the data was grouped. Reading a whole bar back gives an exact frequency, but as soon as a question asks about part of a class, the method assumes the values are spread evenly across that class, which is rarely exactly true. Edexcel writes those parts as estimate rather than work out.

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