Averages from Frequency Tables
Averages from frequency tables is the skill of finding the mode, median, mean and range of discrete data organised into an ungrouped frequency table, where each row lists one specific value, such as a shoe size, alongside how many times it occurred, its frequency, rather than a raw list of individual numbers. Because the table only gives a running count for each value, the frequency column must be used correctly to find the median and mean, unlike grouped data, where values are collected into class intervals and only an estimated mean is possible.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check the table has a value column and a frequency column, and find the total frequency by adding up the frequency column - this is the total number of data items, even though only a few different values are listed.
- To find the mode, read off the value with the single highest frequency - not the frequency number itself.
- To find the median, work out its position using (total frequency + 1) / 2, then count through a running (cumulative) total of the frequencies to see which value that position falls in.
- To find the mean, multiply each value by its frequency to get f x x for every row, add these products together, then divide by the total frequency.
- To find the range, subtract the smallest value that has a non-zero frequency from the largest value that has a non-zero frequency - ignore any value listed with a frequency of 0.
- Add an extra f x x column to the table before totalling, so the multiplication stays organised and arithmetic slips are less likely.
Worked example
The table shows the time, in minutes, spent on homework one evening by 30 pupils: 20 minutes (frequency 4), 30 minutes (frequency 9), 40 minutes (frequency 11), 50 minutes (frequency 6). Find the mode, median, mean and range of the times.
- Total frequency: 4 + 9 + 11 + 6 = 30 pupils.
- Mode: the highest frequency is 11, next to 40 minutes, so the mode is 40 minutes.
- Median: the position is (30 + 1) / 2 = 15.5th, so average the 15th and 16th values. The cumulative frequencies are 4, 13, 24, 30, so positions 14 to 24 are all 40 minutes - both the 15th and 16th values are 40, so the median is 40 minutes.
- Mean: multiply each value by its frequency and add: (20 x 4) + (30 x 9) + (40 x 11) + (50 x 6) = 80 + 270 + 440 + 300 = 1090.
- Divide by the total frequency: 1090 / 30 = 36.3 minutes (1 decimal place).
- Range: the largest value with a non-zero frequency is 50, the smallest is 20, so the range is 50 - 20 = 30 minutes.
Practice questions
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Q1A frequency table shows the shoe sizes of 20 pupils in a class: size 4 (frequency 2), size 5 (frequency 6), size 6 (frequency 7), size 7 (frequency 5). Write down the modal shoe size.Show answer
Answer: Size 6 (it has the highest frequency, 7)
Q2The table shows the number of pets owned by 15 pupils: 0 pets (frequency 3), 1 pet (frequency 5), 2 pets (frequency 4), 3 pets (frequency 3). Find the median number of pets.Show answer
Answer: 1 pet (the 8th value out of 15, ordered; cumulative frequencies are 3, 8, 12, 15, so the 8th value falls in the '1 pet' group)
Q3A netball team recorded the number of goals scored in each of 12 matches: 1 goal (frequency 2), 2 goals (frequency 5), 3 goals (frequency 3), 4 goals (frequency 2). Calculate the mean number of goals per match.Show answer
Answer: 29/12 = 2.42 goals (2 dp) (sum of f x x = 2+10+9+8 = 29, divided by 12)
Q4The table shows the number of rainy days recorded each week for 10 weeks: 2 days (frequency 1), 3 days (frequency 0), 4 days (frequency 4), 5 days (frequency 3), 6 days (frequency 2). Find the range of the number of rainy days.Show answer
Answer: 4 days (largest value with a non-zero frequency is 6, smallest is 2, so 6 - 2 = 4; the frequency-0 row at 3 days is ignored)
Q5In a survey, 25 people were asked how many cups of tea they drink in a day: 0 cups (frequency 4), 1 cup (frequency 9), 2 cups (frequency x), 3 cups (frequency 5). Find the value of x.Show answer
Answer: x = 7 (since 4 + 9 + x + 5 = 25, so 18 + x = 25)
Q6A frequency table for the number of goals scored in 20 hockey matches has its highest frequency, 9, next to the value 2 goals. Write down the mode, and explain why the mode is not 9.Show answer
Answer: Mode = 2 goals. The 9 is the frequency (how many times 2 goals occurred), not a value in the data set itself, so it cannot be the mode.
Q7The table shows the number of correct answers, out of 5, for 16 pupils in a quiz: 2 correct (frequency 3), 3 correct (frequency 6), 4 correct (frequency 4), 5 correct (frequency 3). Calculate the mean number of correct answers.Show answer
Answer: 55/16 = 3.44 (2 dp), or 3.4 (1 dp) (sum of f x x = 6+18+16+15 = 55, divided by 16)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The table shows the number of goals scored by a hockey team in each of 18 matches: 0 goals (frequency 3), 1 goal (frequency 6), 2 goals (frequency 5), 3 goals (frequency 4). (a) Find the median number of goals. (b) Calculate the mean number of goals per match, giving your answer to 1 decimal place.
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The table shows the number of emails received by 24 employees in one day: 5 emails (frequency 4), 6 emails (frequency 9), 7 emails (frequency n), 8 emails (frequency 3). (a) Find the value of n. (b) State the range of the number of emails.
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The frequency table shows the number of visits made to a gym in a month by 40 customers: 2 visits (frequency 6), 4 visits (frequency 14), 6 visits (frequency 12), 8 visits (frequency 8). (a) Write down the modal number of visits. (b) Calculate the mean number of visits per customer, giving your answer to 1 decimal place. (c) A customer claims 'most people visit the gym 8 times a month'. Use your answers to explain whether this claim is correct.
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Free printable worksheet
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This topic is chapter 33 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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