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Cambridge IGCSE0607

Averages and range

International Mathematics 0607 Chapter Notes

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Averages and range
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In this chapter

  1. 1Mean, mode and median of a list
  2. 2Averages from a frequency table
  3. 3Range, quartiles and interquartile range
  4. 4Which average, which spread?
  5. 5Estimating the mean of grouped data
  6. 6The modal class

1. Mean, mode and median of a list

The mean is the total of the values divided by how many there are. The mode is the value that occurs most often. The median is the middle value once the data are in order.

With an even number of values, the median is half-way between the two middle values. The mode is a value from the data, not how many times it occurs.

Mean, mode and median are three different averages. Check which one the question asks for before you start.

mean = (total of the values)/(number of values)

Examiner insight

“Most candidates understood the need to list the values in order, but then did not find the median correctly.” “Some said 21 which was the mode and it is in 6th place in the unordered list.”

Worked example 15 marks

These are the test scores for each of 9 students.

9 4 6 8 9 8 9 7 3

(a) Write down the mode.

(b) Find the mean.

(c) Find the median.

  1. 1

    (a) the most common score is 9 [1]

  2. 2

    (b) mean = total/(number of scores) = 63/9 [1]

  3. 3

    = 7 [1]

  4. 4

    (c) in order: 3, 4, 6, 7, 8, 8, 9, 9, 9 [1]

  5. 5

    the middle (5th) score is 8 [1]

2. Averages from a frequency table

In a frequency table, the mode is the value with the highest frequency. Write the value, not the frequency.

To find the median, add up the frequencies to get the number of values. Count through the table to the middle value, or to the pair of middle values. Use the frequencies, not just the list of values.

To find the mean, multiply each value by its frequency, add these, then divide by the total frequency. On a graphic display calculator, enter the values in one list and the frequencies in another. Choose one-variable statistics and read off the mean.

mean = (sum fx)/(sum f)

Examiner insight

“A few candidates ignored the frequencies and gave an answer of 3.5.” “There were few correct answers seen to this question on finding the mean, with many candidates just finding the mean of the number of petals (ignoring the frequencies) and some finding the mean of the frequencies.”

Worked example 14 marks

The table shows the numbers of letters delivered to each of 90 houses one day.

(a) Find the value of n.

(b) Find the range.

(c) Find the mode.

(d) Find the median.

number of letters0123456
frequency272315108n3
  1. 1

    (a) n = 90 - (27+23+15+10+8+3) = 90 - 86 = 4 [1]

  2. 2

    (b) range = 6 - 0 = 6 [1]

  3. 3

    (c) the highest frequency is 27, so the mode is 0 [1]

  4. 4

    (d) 90 values, so the median is half-way between the 45th and 46th. Running totals: 27, 50, so both are 1 [1]

3. Range, quartiles and interquartile range

The range is the largest value minus the smallest. It is one number, not a pair of values.

Put the data in order. The median splits it into a lower half and an upper half. The lower quartile is the median of the lower half. The upper quartile is the median of the upper half. With an odd number of values, leave the median out of both halves.

The interquartile range is the upper quartile minus the lower quartile. A graphic display calculator gives both quartiles from the one-variable statistics screen.

range = largest - smallest

IQR = Q₃ - Q₁

Examiner insight

“A very common response was to find the range rather than the interquartile range.” “Candidates should be aware that saying ‘24 – 6’ or ‘from 6 to 24’ is not sufficient as the range is a single number, the result of the largest number subtract the smallest.”

Worked example 14 marks

Here are the numbers of goals scored by a team in 8 matches.

12 5 15 8 3 13 6 9

Find the lower quartile, the upper quartile and the interquartile range.

  1. 1

    in order: 3, 5, 6, 8, 9, 12, 13, 15 [1]

  2. 2

    lower half: 3, 5, 6, 8, so Q₁ = (5+6)/2 = 5.5 [1]

  3. 3

    upper half: 9, 12, 13, 15, so Q₃ = (12+13)/2 = 12.5 [1]

  4. 4

    IQR = 12.5 - 5.5 = 7 [1]

4. Which average, which spread?

The mean, the median and the mode are averages. The range and the interquartile range measure how spread out the data are.

An average is chosen to suit the data. If one value is much larger than the others, the median is the suitable average. If the mode is the highest value in the data, it is not a suitable average. Answer about the data in the question, not with a definition.

To compare two sets of data, compare an average and compare a spread. A smaller range means the values are more consistent.

Examiner insight

“Many candidates simply described the mode and stated that the mean is a better average.” “The important idea was that, in this case, the mode happened to be the highest score and therefore unsuitable as a measure of average.”

Worked example 14 marks

This is a list of the times, in seconds, for 8 boys to swim 50 metres.

43.9 30.6 31.3 49.8 46.4 39.7 52.3 44.8

(a) Work out the mean.

(b) Work out the range.

(c) The table shows a summary of the times, in seconds, for 8 girls to swim 50 metres. Make two comparisons between the swimming times of the boys and of the girls.

mean37.25
range26.3
  1. 1

    (a) (43.9+30.6+31.3+49.8+46.4+39.7+52.3+44.8)/8 = 338.8/8 = 42.35 [1]

  2. 2

    (b) 52.3 - 30.6 = 21.7 [1]

  3. 3

    (c) the girls' average time is lower: 37.25 is less than 42.35 [1]

  4. 4

    the boys' times are more consistent: their range, 21.7, is smaller than 26.3 [1]

5. Estimating the mean of grouped data

Grouped data do not show the exact values, so the mean can only be estimated. Use the mid-value of each class: the number half-way between its two ends.

Multiply each mid-value by its frequency, add these, and divide by the total frequency. The answer is an estimate.

On a graphic display calculator, enter the mid-values in one list and the frequencies in the other. Then read off the mean.

estimated mean = (sum f × mid-value)/(sum f)

Examiner insight

“Of those who made a reasonable attempt, a number used the end value in each group rather than the middle value.” “Some candidates found the mean of the frequencies whilst others also used the width of the interval rather than the midpoint.”

Worked example 12 marks

The table shows the times taken by 800 people to complete a task. Calculate an estimate of the mean time.

time (t seconds)0 < t leqslant 2525 < t leqslant 3535 < t leqslant 4040 < t leqslant 5050 < t leqslant 70
frequency1590280275140
  1. 1

    mid-values: 12.5, 30, 37.5, 45, 60

  2. 2

    mean ≈ (15×12.5 + 90×30 + 280×37.5 + 275×45 + 140×60)/800 [1]

  3. 3

    = (34 162.5)/800 = 42.7 s [1]

6. The modal class

In a grouped frequency table, the modal class is the class with the highest frequency.

Give the class itself, such as 16 to 20, not its frequency. Give one class, not the whole range of the table.

To find the group that contains the median, add up the frequencies until you reach the middle value.

Examiner insight

“Although many candidates found the correct modal class, others wrote 41 to 60 or 0 to 100.”

Worked example 12 marks

The table shows the marks scored by 50 students in a test. Find(i) the modal class,(ii) the group that contains the median.

mark1 to 56 to 1011 to 1516 to 2021 to 2526 to 3031 to 35
frequency137129117
  1. 1

    (i) the highest frequency is 12, so the modal class is 16 to 20 [1]

  2. 2

    (ii) 50 values, so the median is between the 25th and 26th. Running totals: 1, 4, 11, 23, 32. The group is 21 to 25 [1]

Recap

  • Mean = total of the values ÷ number of values. Mode = most common value. Median = middle value in order.
  • Even number of values: the median is half-way between the two middle values.
  • Frequency table: mode is the value with the highest frequency. Mean = sum fx ÷ sum f.
  • Range = largest − smallest, a single number. IQR = upper quartile − lower quartile.
  • Quartiles are the medians of the lower and upper halves of the ordered data.
  • Use the median if one value is much larger than the others. A smaller range means more consistent.
  • Extended: estimate the mean of grouped data using the mid-value of each class.
  • Extended: the modal class is the class with the highest frequency.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write down the mode of these numbers: 4, 7, 7, 2, 9, 4, 7, 5.1 mark
  2. Do not use a calculator. Find the median of these numbers: 12, 7, 15, 9, 20, 11.2 marks
  3. Do not use a calculator. The mean of five numbers is 8. Four of the numbers are 5, 9, 12 and 11. Find the fifth number.2 marks
  4. The table shows the number of goals scored in each of 20 matches. (a) Find the median. (b) Calculate the mean. | goals | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | --- | | frequency | 4 | 6 | 5 | 3 | 2 |4 marks
  5. Do not use a calculator. Here are 9 numbers: 4, 8, 10, 3, 12, 7, 15, 9, 5. Find the interquartile range.3 marks
  6. The prices of five houses in a street, in thousands of pounds, are 210, 225, 230, 235 and 1200. State whether the mean or the median is the more suitable average. Give a reason.2 marks
  7. Do not use a calculator. The table shows the times that 20 students take to travel to school. Estimate the mean time. | time (t minutes) | 0 < t leqslant 10 | 10 < t leqslant 20 | 20 < t leqslant 40 | | --- | --- | --- | --- | | frequency | 4 | 6 | 10 |3 marks
  8. The table shows the lengths of 40 leaves. (a) Write down the modal class. (b) Which class contains the median? | length (l cm) | 0 < l leqslant 4 | 4 < l leqslant 8 | 8 < l leqslant 12 | 12 < l leqslant 16 | 16 < l leqslant 20 | | --- | --- | --- | --- | --- | --- | | frequency | 5 | 14 | 9 | 8 | 4 |2 marks
  9. Do not use a calculator. A librarian records the number of books borrowed by each member of a book club in one year. The estimate of the mean is exactly 13 books. (a) Find the value of k. (b) Write down the modal class. (c) Find the group that contains the median. | number of books | 1 to 5 | 6 to 10 | 11 to 15 | 16 to 30 | | --- | --- | --- | --- | --- | | frequency | 4 | k | 9 | 10 |6 marks
  10. A courier records the mass, m kg, of each of the 80 parcels in van A. (a) Calculate an estimate of the mean mass of a parcel in van A. (b) Van B carries 120 parcels. Their mean mass is 2.5 kg. Calculate an estimate of the mean mass of all 200 parcels. (c) One parcel in van A has a mass of exactly 2 kg. It was counted in the class 2 < m leqslant 5 by mistake. The table is corrected. Calculate the new estimate of the mean mass for van A. | mass (m kg) | 0 < m leqslant 1 | 1 < m leqslant 2 | 2 < m leqslant 5 | 5 < m leqslant 8 | 8 < m leqslant 12 | | --- | --- | --- | --- | --- | --- | | frequency | 14 | 22 | 30 | 10 | 4 |7 marks

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