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Cambridge IGCSE0580

Averages and range

Mathematics 0580 Chapter Notes

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Averages and range
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In this chapter

  1. 1Mean, median, mode and range of a list
  2. 2Mode and median from a frequency table
  3. 3Mean from a frequency table
  4. 4Choosing an average
  5. 5Working backwards from an average
  6. 6Quartiles and interquartile range
  7. 7Grouped data: modal class and estimated mean

1. Mean, median, mode and range of a list

The mean is the sum of the values divided by the number of values. The median is the middle value once the data are in order. The mode is the value that occurs most often. The range is the largest value minus the smallest.

With an even number of values, the median is halfway between the two middle values. Always put the data in order first.

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

range = largest value - smallest value

Examiner insight

“Writing + signs between the digits then ÷ 9 is not enough for the method mark without brackets around the numbers to be added.” “Another error was to round to the whole number, 35, even though this was an exact answer.”

Worked example 14 marks

Raj records the temperature, in °C, at midday for each of 8 days. These are the results.

1, 10, 8, −3, −2, 10, 7, 9

(a) Calculate the mean.

(b) Find the median.

  1. 1

    (a) sum = 1 + 10 + 8 + (-3) + (-2) + 10 + 7 + 9 = 40 [1]

  2. 2

    mean = 40 ÷ 8 = 5 °C [1]

  3. 3

    (b) in order: -3, -2, 1, 7, 8, 9, 10, 10 [1]

  4. 4

    median = (7 + 8)/2 = 7.5 °C [1]

2. Mode and median from a frequency table

In a frequency table the top row gives the values and the bottom row says how often each occurs.

The mode is the value with the highest frequency, not the frequency itself. For the median, find the position of the middle value, then add the frequencies in order until you reach that position.

Examiner insight

“The mode is 4 visits not the frequency of 7.” “A common incorrect answer was 8.5, the result of finding the median of the list of reaction times without considering the number of students with each time.”

Worked example 12 marks

A librarian records the number of books borrowed by each of 40 students. The frequency table shows the results.

(a) Find the mode.

(b) Find the median.

number of books012345
frequency51410812
  1. 1

    (a) the highest frequency is 14, for 1 book: mode = 1 [1]

  2. 2

    (b) the median lies between the 20th and 21st of 40 values; adding the frequencies in order gives 5, 19, 29, so both are 2: median = 2 [1]

3. Mean from a frequency table

Multiply each value by its frequency. Add these products. Divide by the total frequency.

Do not divide by the number of columns. A quick check: the mean must lie between the smallest and largest values.

mean = (total of (value × frequency))/(total frequency)

Examiner insight

“Each number of visits must be multiplied by its frequency and all of these products added together, and finally, the total is divided by 40.”

Worked example 13 marks

A librarian records the number of books borrowed by each of 40 students. The frequency table shows the results.

Calculate the mean.

number of books012345
frequency51410812
  1. 1

    products: 0 × 5, 1 × 14, 2 × 10, 3 × 8, 4 × 1, 5 × 2 [1]

  2. 2

    total = 0 + 14 + 20 + 24 + 4 + 10 = 72 [1]

  3. 3

    mean = 72 ÷ 40 = 1.8 [1]

4. Choosing an average

The mean uses every value, so one value much larger than the rest pulls it up. The median is the middle value and is not pulled in this way. The mode is the most common value.

When a question asks why an average is not suitable, name the value that causes the problem and say how it affects the average.

Examiner insight

“Only a very small proportion of candidates knew why the mean was not a suitable average to use in this situation, i.e. because the 96 kg parcel was an extreme value so would distort the mean.”

Worked example 13 marks

These are the wages of 8 office workers, in dollars.

220, 210, 850, 265, 245, 230, 245, 235

(a) Work out the median.

(b) Give a reason why the mean may not be a suitable average to represent the data.

  1. 1

    (a) in order: 210, 220, 230, 235, 245, 245, 265, 850 [1]

  2. 2

    median = (235 + 245)/2 = 240 dollars [1]

  3. 3

    (b) 850 is much larger than the other values, which affects the mean [1]

5. Working backwards from an average

A mean tells you the total: total = mean × number of values. When a value is added, find the new total and subtract the old one.

A median, a mode and a range each fix where values sit in order. Write the values in order and test each clue.

total = mean × number of values

Examiner insight

“Some candidates did not take into account that all values had to be different, for example [–3], 6, 8, 8, 11.” “Most candidates used trial and improvement to find answers that fulfilled some of the three criteria.”

Worked example 13 marks

A group of 4 children have a mean age of 11. A fifth child joins the group, and the mean age is now 10. Calculate the age of the fifth child.

  1. 1

    total of 4 ages = 4 × 11 = 44 [1]

  2. 2

    total of 5 ages = 5 × 10 = 50 [1]

  3. 3

    fifth child = 50 - 44 = 6 years [1]

6. Quartiles and interquartile range

Put the data in order. The lower quartile is a quarter of the way along the list and the upper quartile is three quarters of the way along. The interquartile range is the upper quartile minus the lower quartile.

Find each position using n + 1, where n is the number of values. If a position falls halfway between two values, take the value halfway between them.

lower quartile at position (n+1)/4 upper quartile at position (3(n+1))/4

IQR = upper quartile - lower quartile

Examiner insight

“Candidates who chose to calculate the positions of the upper quartile and lower quartile were the most successful, provided they remembered to use n + 1 rather than n in their calculations.” “Part (b) was answered more successfully than part (a), although a small number of candidates mixed up range and interquartile range.”

Worked example 14 marks

Do not use a calculator. Here are 11 scores.

12, 3, 17, 8, 20, 5, 15, 9, 13, 18, 6

Find the interquartile range.

  1. 1

    in order: 3, 5, 6, 8, 9, 12, 13, 15, 17, 18, 20 [1]

  2. 2

    lower quartile is at position (11 + 1)/4 = 3, so it is 6 [1]

  3. 3

    upper quartile is at position 3 × 3 = 9, so it is 17 [1]

  4. 4

    IQR = 17 - 6 = 11 [1]

7. Grouped data: modal class and estimated mean

In grouped data you know the class each value falls in, not the value itself. The modal class is the class with the highest frequency.

To estimate the mean, use the midpoint of each class as its value. Multiply each midpoint by its frequency, add, and divide by the total frequency. It is only an estimate because the true values are not known.

midpoint = (lower limit + upper limit)/2

estimated mean = (total of (midpoint × frequency))/(total frequency)

Examiner insight

“A few candidates used the class width of the interval instead of midpoints.” “Others wrote down the mid-points correctly but did not multiply by the frequencies.”

Worked example 15 marks

The table shows the marks of 100 students in a test.

(a) Write down the modal class.

(b) Calculate an estimate of the mean.

mark0 < x ≤ 2020 < x ≤ 4040 < x ≤ 6060 < x ≤ 8080 < x ≤ 100
frequency420313411
  1. 1

    (a) the highest frequency is 34: modal class is 60 < x ≤ 80 [1]

  2. 2

    (b) midpoints: 10, 30, 50, 70, 90 [1]

  3. 3

    4 × 10 + 20 × 30 + 31 × 50 + 34 × 70 + 11 × 90 = 5560 [1]

  4. 4

    5560 ÷ 100 [1]

  5. 5

    = 55.6 [1]

Recap

  • Mean = sum of values ÷ number of values. Median = middle value in order. Mode = most common value. Range = largest minus smallest.
  • Frequency table: the mode is the value with the highest frequency. Mean = total of value × frequency ÷ total frequency.
  • The mean is pulled by a value much larger than the rest. The median is not.
  • Working backwards: total = mean × number of values.
  • Extended: lower quartile at position (n + 1) ÷ 4, upper quartile at 3(n + 1) ÷ 4. IQR = upper quartile minus lower quartile.
  • Extended: grouped data. Modal class = highest frequency. Estimated mean uses midpoints.

End-of-chapter exercise

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

  1. Do not use a calculator. Find the mode of 4, 7, 7, 3, 9, 4, 7, 2.1 mark
  2. Do not use a calculator. The temperatures at midnight on five nights, in °C, were 3, −4, 0, −7, 5. Find the range.2 marks
  3. Do not use a calculator. Find the median of 12, 5, 9, 20, 7, 15.2 marks
  4. The masses, in kg, of five parcels are 2.4, 3.1, 1.9, 2.8 and 2.3. Calculate the mean mass.2 marks
  5. A coach records the number of goals scored by her team in each of 30 matches. Find the mode and the median. | goals | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | --- | | frequency | 6 | 9 | 8 | 5 | 2 |3 marks
  6. A survey asks 30 families how many pets they have. Calculate the mean number of pets. | pets | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | | frequency | 7 | 12 | 9 | 2 |3 marks
  7. Do not use a calculator. The prices of five houses in a street, in thousands of dollars, are 210, 225, 230, 240 and 1095. The seller says the mean price is the best average to show a typical house. Work out the mean and the median and say whether the seller is right.3 marks
  8. Do not use a calculator. The mean of five numbers is 12. Four of the numbers are 7, 10, 13 and 17. Find the fifth number.3 marks
  9. Do not use a calculator. A botanist measures the heights, in cm, of six seedlings: 9, 4, 12, 6, 15, 10. She then measures three more seedlings. Their heights, in cm, are x, x and 16. The mean height of all nine seedlings is 10 cm. Find the interquartile range of the nine heights.6 marks
  10. A bus company records how late each of 50 buses is on Monday. (a) Find the value of k. (b) Write down the modal class. (c) Calculate an estimate of the mean for Monday. (d) On Tuesday the company records 30 buses. An estimate of their mean is 15.5 minutes. Calculate an estimate of the mean for all 80 buses. | time late (t minutes) | 0 < t ≤ 5 | 5 < t ≤ 10 | 10 < t ≤ 20 | 20 < t ≤ 40 | | --- | --- | --- | --- | --- | | frequency | 8 | k | 14 | 6 |8 marks

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