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Cambridge IGCSE0607

Averages on a calculator

International Mathematics 0607 Chapter Notes

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Averages on a calculator
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In this chapter

  1. 1Entering a list of data
  2. 2Median and quartiles
  3. 3Mean from a frequency table
  4. 4Mean for grouped data

1. Entering a list of data

A graphic display calculator finds the mean, median and quartiles of a list of discrete data in one go.

Enter each value into a list. Choose the one-variable statistics. Then read off the mean, the median and the lower and upper quartiles from the screen.

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

Examiner insight

“It was expected that the candidates would enter the data in their graphical display calculator and read off the answers, but it appeared that most did not do this.” “The graphics calculator can be used for the median and quartiles or a simple ordered list would have helped.”

Worked example 11 mark

The table shows the marks of 10 students in a French exam and a Spanish exam. Find the mean French mark.

French mark (x)18223136424855636976
Spanish mark (y)34424355485859687279
  1. 1

    enter the ten French marks in a list, then read off the mean: barx = 460/10 = 46 [1]

2. Median and quartiles

The median is the middle value of the data when they are in order. With an even number of values it is halfway between the two middle ones, so it need not be one of the items of data.

The lower quartile is halfway from the minimum value to the median in the ordered data. The upper quartile is halfway from the median to the maximum value.

The calculator orders the data for you and gives the median and both quartiles. Read the right line from the screen.

Examiner insight

“The median was often correctly found whilst a few candidates gave a student as the answer and others appeared to think that the answer must be one of the items of data.” “In part (ii), some confused mean with median and in part (iii), a number of candidates found the middle value of the given list rather than an ordered list.”

Worked example 13 marks

The numbers of goals scored by a team in 10 matches are

7, 12, 4, 15, 8, 15, 13, 7, 20, 10.

Use your calculator to find the median, the lower quartile and the upper quartile.

  1. 1

    enter the ten values in a list and run the one-variable statistics

  2. 2

    median = 11 (halfway between 10 and 12 in the ordered list) [1]

  3. 3

    lower quartile = 7 [1]

  4. 4

    upper quartile = 15 [1]

3. Mean from a frequency table

A frequency table lists each value once, with how many times it occurs. Put the values in one list and the frequencies in a second list.

Tell the calculator to use the frequency list. The mean is then the total of value × frequency divided by the total frequency.

The median and the quartiles must take the frequencies into account too. The frequency list does this for you.

mean = (sum fx)/(sum f)

Examiner insight

“Many added 1 + 2 + …+ 6 and divided the total by 6. Others added the frequencies and divided by 6.” “Most candidates just added up the frequencies and divided by 7.”

Worked example 13 marks

The table shows the number of goals scored in each of 20 matches.

Find the mean number of goals.

Goals (x)01234
Frequency (f)25832
  1. 1

    values in one list, frequencies in the second list

  2. 2

    sum fx = 0 × 2 + 1 × 5 + 2 × 8 + 3 × 3 + 4 × 2 = 38 [1]

  3. 3

    divide by sum f = 20 [1]

  4. 4

    mean = 38/20 = 1.9 [1]

4. Mean for grouped data

Grouped data give classes, not exact values. Use the middle value of each class as x and the class frequency as f.

Enter the middle values in one list and the frequencies in the other. The calculator gives the mean. Write down the mid-interval values as your working.

The result is an estimate, because the middle value stands in for every value in the class.

middle value = (lower limit + upper limit)/2

mean = (sum fx)/(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.” “Weaker candidates often added the mid-interval values and divided by 6.”

Worked example 14 marks

The table shows the times, t minutes, taken by 20 students to finish a puzzle.

Calculate an estimate of the mean time.

Time (t minutes)0 < t ≤ 1010 < t ≤ 2020 < t ≤ 3030 < t ≤ 40
Frequency3584
  1. 1

    middle values 5, 15, 25, 35 [1]

  2. 2

    sum fx = 15 + 75 + 200 + 140 = 430 [1]

  3. 3

    sum f = 20 [1]

  4. 4

    mean = 430/20 = 21.5 minutes [1]

Recap

  • Enter the data in a list and use the one-variable statistics.
  • The screen gives the mean, median, lower quartile and upper quartile.
  • The median is the middle value of the ordered data.
  • With a frequency table, use a second list for the frequencies.
  • Mean = sum fx ÷ sum f.
  • For grouped data, use the middle value of each class.
  • Show the mid-interval values as working for a grouped mean.

End-of-chapter exercise

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

  1. State which average is found by taking the middle value of the data in order.1 mark
  2. The ages of seven players are 23, 18, 25, 19, 30, 22 and 21. Use your calculator to find the mean age. Give your answer to 3 significant figures.2 marks
  3. Find the median of 14, 9, 21, 12, 17, 8, 15.2 marks
  4. Find the median of 6, 3, 9, 4, 8, 12.2 marks
  5. Ten students scored these marks in a test: 12, 7, 15, 9, 20, 11, 14, 8, 18, 10. Use your calculator to find the lower quartile and the upper quartile.3 marks
  6. The table shows the scores of 20 players in a game. Find the mean score. | Score (x) | 1 | 2 | 3 | 4 | 5 | | --- | --- | --- | --- | --- | --- | | Frequency (f) | 3 | 6 | 8 | 2 | 1 |3 marks
  7. The table shows the number of goals scored in each of 28 matches. Find the median number of goals. | Goals (x) | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | --- | | Frequency (f) | 5 | 9 | 7 | 4 | 3 |3 marks
  8. The table shows the times, t minutes, that 30 people waited at a clinic. Calculate an estimate of the mean waiting time. Give your answer to 3 significant figures. | Time (t minutes) | 0 < t ≤ 10 | 10 < t ≤ 20 | 20 < t ≤ 30 | 30 < t ≤ 40 | | --- | --- | --- | --- | --- | | Frequency | 6 | 11 | 9 | 4 |4 marks
  9. A courier weighs 40 parcels. The table shows the masses, m kg. (a) Calculate an estimate of the mean mass. (b) Ten more parcels arrive. Each has a mass in the class 6 < m ≤ 10. Calculate an estimate of the mean mass of all 50 parcels. | Mass (m kg) | 0 < m ≤ 2 | 2 < m ≤ 4 | 4 < m ≤ 6 | 6 < m ≤ 10 | | --- | --- | --- | --- | --- | | Frequency | 8 | 14 | 10 | 8 |5 marks
  10. A hostel records the number of nights each of 40 guests stays. (a) Use your calculator to find the mean, the median, the lower quartile and the upper quartile of the number of nights. (b) Three more guests each stay 14 nights. Find the new median and the new mean. Give the mean to 3 significant figures. | Nights (x) | 1 | 2 | 3 | 4 | 5 | 7 | | --- | --- | --- | --- | --- | --- | --- | | Frequency (f) | 6 | 9 | 11 | 7 | 4 | 3 |6 marks

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