Cumulative frequency: worked examples and method

Median

The total frequency is 24. The curve reads 20 at cumulative frequency 12. The median is 20.

Lower quartile

A quarter of the way up the curve is the lower quartile.

Upper quartile

Three quarters of the way up is the upper quartile.

Compare

Set A has median 20 and set B has median 28. The higher median is B, so set B is typically higher. To compare how spread out the sets are, use the interquartile range, which is the upper quartile minus the lower quartile.

Axes

Cumulative frequency is how many so far. The other axis is the value you report.

Half, quarter, three quarters

Median uses half the total. Quartiles use one quarter and three quarters.

The running total

Cumulative frequency only grows or stays the same. You never subtract a class from the running total. The first cumulative frequency equals the first frequency, because nothing came before it. After that, add the next frequency on. A table with frequencies 6, 8 and 16 has cumulative frequencies 6, 14 and 30.

The last number is n, the total. Median and quartile positions are fractions of this n, so a wrong running total moves every answer. If the question gives the cumulative frequencies ready-made, check that they increase and that the last one is the total they mention. "44 results" and a final cumulative frequency of 44 are the same fact said twice.

Method: 1. Start the running total with the frequency of the first class. 2. Each later entry is the previous cumulative frequency plus the new class. 3. Check the running totals never decrease. 4. Use the final running total as n when you place the median and quartiles.

Plot at the upper boundary

A class written as 20 < t ≤ 30 has upper boundary 30. Everyone in that class took 30 minutes or less, and the cumulative frequency says how many people that is. The point is (30, cumulative frequency). Plotting at the midpoint 25 would be the frequency-polygon habit, and it shifts the whole curve left.

The curve starts on the axis at the lowest boundary if you want a closed shape, with cumulative frequency 0. That starting point is not a class of people. When you read a median, you use the curve and the positions. You do not add the starting zero into n.

Method: 1. Identify the upper end of each class. 2. Plot the cumulative frequency at that upper end. 3. Do not plot it at the midpoint. 4. Close the curve with a point at cumulative frequency 0, and do not count it as another class.

Median and quartiles

For a cumulative frequency graph the median position is n/2, the lower quartile is n/4 and the upper quartile is 3n/4. For 80 people those positions are 40, 20 and 60. The +1 rule belongs to a list of separate values, where you are picking an item. Here you are reading a scale that already treats the data as continuous, so you use n itself.

If the position equals a cumulative frequency in the table, stop. The quartile is the upper boundary that carries that cumulative frequency. No interpolation, and no guessing between grid lines. Position 20 landing on cumulative frequency 20 at the upper boundary 25 means the lower quartile is 25. That is exact.

Method: 1. Read n from the last cumulative frequency. 2. Median position n/2. Lower quartile n/4. Upper quartile 3n/4. 3. If that position matches a cumulative frequency exactly, read its upper boundary. 4. If the position lands on a boundary, use that boundary as the quartile.

Interpolation and the IQR

If the position falls between two cumulative frequencies, the value falls between their upper boundaries. Suppose position 22 sits between cumulative frequency 14 at boundary 20 and cumulative frequency 30 at boundary 30. The fraction along the class is (22-14)/(30-14) = 1/2. The class width is 10, so you add 5 to 20. The median is 25. The arithmetic has to divide evenly. If it does not, check the subtraction.

The interquartile range is the upper quartile minus the lower quartile. It is the spread of the middle 50 percent. A large IQR means that middle half is widely spread. Subtracting the wrong way, or doing upper divided by lower, is not a spread. Once both quartiles are known, whether by an exact hit or by interpolation, the subtraction is the last line.

Method: 1. Find which two cumulative frequencies trap the position. 2. Work out how far the position sits through the class, as a fraction of that class. 3. Multiply that fraction by the class width and add it to the previous upper boundary. 4. IQR = upper quartile − lower quartile.

Completing the cumulative frequency table

Exam questions usually start with a grouped frequency table and one mark for "Complete the cumulative frequency table". Add each frequency to the total so far. Frequencies of 6, 14, 20, 14 and 6 give cumulative frequencies of 6, 20, 40, 54 and 60. The last one must equal the total number of results, which is a built-in check.

The cumulative frequency column is labelled with the upper boundary of each class, such as t ≤ 40. The first point on the graph is at the lowest boundary with cumulative frequency 0. Each point is (upper boundary, cumulative frequency). Write them as coordinates if asked, for example (40, 20). Do not use the class frequency as a coordinate.

Method: 1. Copy the first frequency, then add the next frequency to the running total. 2. Continue down the table, writing each running total. 3. Check the last cumulative frequency equals the total number of results. 4. Write each point as (upper boundary, cumulative frequency), with (lowest boundary, 0) first.

Tip: If the last total does not match the number of results, find the slip before you plot anything.

Reading values by interpolating between points

To estimate a median or quartile, first find its position: n/2 for the median, n/4 for the lower quartile and 3n/4 for the upper quartile. On the graph you go across at that height and down to the axis. In this lesson the points are joined by straight lines, so you can calculate the same reading by proportion.

Find the two table points either side of the position, say (x_1, c_1) and (x_2, c_2). The estimate is x_1 + dfracposition - c_1c_2 - c_1 × (x_2 - x_1). For (40, 20), (60, 40) and position 30 this is 40 + 10/20 × 20 = 50. The answer is a value on the horizontal axis, not a frequency, so it should carry the unit of the data, such as minutes or kilograms, and it must lie inside the class you used.

Method: 1. Work out the position from n. 2. Find the two points in the table that the position sits between. 3. Work out what fraction of the way along the position is, using the cumulative frequencies. 4. Move that fraction of the class width from the lower boundary, and write the value with its unit.

Tip: The median is a value on the horizontal axis. If your answer is the same as n ÷ 2, you have probably read the wrong axis.

How many above, below or between a value

Sometimes the question gives a value and asks how many results are bigger or smaller. Now you start from the horizontal axis. Find the cumulative frequency at that value, again by interpolating between two table points. The result is the number of results that are less than or equal to the value.

For "more than" questions, subtract that from the total. For "between" questions, subtract the two cumulative frequencies. A percentage claim, such as "at least 60% earn £385 or less", needs the cumulative frequency at 385 divided by the total. Compare it with the claim and write a conclusion using the numbers you found.

Method: 1. Find the two table points either side of the given value. 2. Work out the cumulative frequency at that value by proportion. 3. Subtract from the total for "more than", or subtract two readings for "between". 4. For a percentage, divide by the total, multiply by 100, and compare with the claim.

Comparing groups and using percentiles (Higher)

To compare two cumulative frequency graphs, find the median and the interquartile range of each. A higher median means a higher average. A smaller interquartile range means more consistent values. Put the figures in a sentence with the context, as you do for box plots, because a bare "A is better" earns no marks.

A percentile marks a position in the same way as a quartile. The 90th percentile is at position 0.9 × n. It is the value that only 10% of the results exceed. Questions may ask for the value that the top 10% of scores are above, or the mark needed to be in the top tenth. Use the same interpolation, with the new position.

Method: 1. Work out the median and the lower and upper quartiles for each group. 2. Subtract the quartiles to get the interquartile range for each group. 3. Write one sentence comparing the medians and one comparing the interquartile ranges. 4. For a percentile, find p/100 × n and interpolate as before.