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9a · E9.1–E9.5

Averages, charts and scatter diagrams

Topic 9 Statistics · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Classify and tabulate data (tally tables, two-way tables); read and compare tables and diagrams, and know the limits of the conclusions.
  • Calculate the mean, median, mode and range from a list or an ungrouped frequency table, and choose the right average.
  • Find quartiles and the interquartile range, estimate the mean of grouped data and find the modal class EXTENDED.
  • Draw and interpret bar charts (including stacked and dual), pie charts, pictograms, stem-and-leaf diagrams and frequency distributions.
  • Draw and interpret scatter diagrams, describe correlation, and draw and use a line of best fit.

📚The mathematics

Averages and spread

mean \( = \dfrac{\text{total}}{\text{number of values}} \)
median: middle value in order, position \( \tfrac{n + 1}{2} \)
mode: most common value
range \( = \) largest \( - \) smallest

From a frequency table, the mean is \( \dfrac{\sum fx}{\sum f} \) — divide by the total frequency, not by the number of rows. The median is found by counting through the cumulative frequencies. The median is the better average when there are extreme values (outliers); the mode suits data that are not numbers.

EXTENDED The lower and upper quartiles are the medians of the bottom and top halves (positions \( \tfrac{n + 1}{4} \) and \( \tfrac{3(n + 1)}{4} \)); the interquartile range \( = \) UQ \( - \) LQ measures the spread of the middle half and ignores outliers. For grouped data, use each class’s midpoint as \(x\): the result is an estimate of the mean, because the exact values are unknown. The modal class is the class with the highest frequency.

Charts

In a pie chart, each sector angle is \( \tfrac{\text{frequency}}{\text{total}} \times 360^{\circ} \). A stem-and-leaf diagram must be ordered and have a key; it keeps every data value, so medians and quartiles can be read straight off it.

An ordered stem-and-leaf diagram of 11 values with key 1 | 2 means 12: stem 1 has leaves 2, 5, 8; stem 2 has 0, 3, 3, 7, 9; stem 3 has 1, 4; stem 4 has 2. The lower quartile 18, the median 23 and the upper quartile 31 are highlighted.
Practice question 4: 11 values, so the median is the 6th (23), the quartiles the 3rd (18) and 9th (31).
A pie chart of how 72 students travel to school: walk 18 students, 90 degrees; bus 30 students, 150 degrees; car 16 students, 80 degrees; bicycle 8 students, 40 degrees.
72 students, so each student is \( 360 \div 72 = 5^{\circ} \). Walk: \( 18 \times 5 = 90^{\circ} \).

Scatter diagrams

Positive correlation: as one variable increases, so does the other; negative: one goes up as the other goes down; no correlation: no pattern. A line of best fit is a single ruled straight line drawn by eye, following the trend across the whole data set with roughly as many points either side; it need not pass through the origin or any particular point. Use it to estimate within the range of the data — predictions far outside that range are unreliable.

A scatter diagram of hours of revision against test score for 10 students, plotted as crosses, showing positive correlation. A ruled line of best fit runs through the middle of the points, and dashed lines show that 6 hours of revision gives an estimated score of about 68.
Positive correlation. From the line of best fit, 6 hours of revision suggests a score of about 68. (Illustrative data.)

✏️Worked example

(a) (Non-calculator.) The table shows the number of pets owned by 30 students: 0 pets 6, 1 pet 9, 2 pets 7, 3 pets 5, 4 pets 3. Find the mean, median, mode and range. [5] (b) (Calculator.) EXTENDED The times \(t\) minutes of 40 journeys are: \( 0 < t \le 10 \): 4; \( 10 < t \le 20 \): 11; \( 20 < t \le 30 \): 15; \( 30 < t \le 50 \): 10. Calculate an estimate of the mean, and state the modal class. [4]

(a) \( \sum fx = 0 + 9 + 14 + 15 + 12 = 50 \), so the mean \( = \tfrac{50}{30} = 1.67 \) (3 s.f.). The median is between the 15th and 16th values; the cumulative frequencies are 6, 15, 22, …, so the 15th value is 1 and the 16th is 2: median \( = 1.5 \). Mode \( = 1 \). Range \( = 4 - 0 = 4 \).

(b) Midpoints 5, 15, 25, 40: \( \dfrac{5(4) + 15(11) + 25(15) + 40(10)}{40} = \dfrac{960}{40} = 24 \) minutes. Modal class \( 20 < t \le 30 \).

Check it. Every mean must lie between the smallest and largest values: 1.67 is between 0 and 4, and 24 is between 0 and 50 ✓.
Dividing by 5. In (a), \( 50 \div 5 = 10 \) pets on average is absurd: divide by the number of students (30), not the number of rows. In (b), note the last class is 20 wide, so its midpoint is 40, not 35.

📝Practise

Questions in the style of the current papers. EXTENDED marks Extended-only content.

1. (Non-calculator.) Find the mean, median and range of 7, 3, 9, 4, 12. [3]
Mean \( 35 \div 5 = 7 \). In order 3, 4, 7, 9, 12: median 7. Range \( 12 - 3 = 9 \).
2. (Non-calculator.) The mean of six numbers is 8. One number is removed and the mean of the other five is 7.4. Find the number removed. [3]
Totals: \( 6 \times 8 = 48 \) and \( 5 \times 7.4 = 37 \). Removed number \( = 48 - 37 = 11 \).
3. (Non-calculator.) In a pie chart of 72 students, 18 walk to school. Find the angle of the “walk” sector. [1]
\( \tfrac{18}{72} \times 360 = 90^{\circ} \).
4. (Non-calculator.) Using the stem-and-leaf diagram above, find the median, mode and range. EXTENDED Also find the interquartile range. [4]
Median (6th value) 23; mode 23; range \( 42 - 12 = 30 \). LQ (3rd value) 18, UQ (9th value) 31, so IQR \( = 13 \).
5. (Calculator.) EXTENDED Heights \(h\) cm of 25 plants: \( 140 < h \le 150 \): 5; \( 150 < h \le 160 \): 12; \( 160 < h \le 170 \): 8. Calculate an estimate of the mean height. [3]
\( \dfrac{145(5) + 155(12) + 165(8)}{25} = \dfrac{3905}{25} = 156.2 \) cm.
6. (Non-calculator.) Using the scatter diagram above, describe the correlation, and explain why using the line to predict the score for 20 hours of revision would be unreliable. [2]
Positive correlation. 20 hours is far outside the data (0 to 10 hours), so the trend may not continue — and a score cannot exceed 100.
7. (Non-calculator.) The salaries in a small company are $2000, $2100, $2200, $2300 and $12 000 a month. Which average best represents a typical salary? Give a reason. [2]
The median ($2200): the mean ($4120) is pulled up by the one very large salary, so it is higher than four of the five salaries.

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • Corbettmaths — averages from frequency tables and estimating the mean of grouped data
  • Desmos — enter a table of points to plot a scatter diagram