Averages from a Frequency Table
Averages from a frequency table means calculating the mean, median, mode and range of data that has already been grouped into a table showing each value and how often it occurs. The mode is the value with the highest frequency, the median is found using the running total (cumulative frequency), and the mean is calculated as (sum of frequency x value) divided by the total frequency. This is a common GCSE Statistics topic tested with discrete data such as shoe sizes or number of siblings.
Before you start
Make sure you're comfortable with these topics first:
Method
- Add a third column to the table showing frequency x value (f x x) for each row.
- For the mean, sum the frequency x value column, then divide by the total frequency (sum of the frequency column).
- For the mode, read off the value with the highest frequency directly from the table.
- For the median, add a running total (cumulative frequency) column, then find which value contains the ((total frequency + 1)/2)th data item.
- For the range, subtract the smallest value in the table from the largest value, using only values that actually have a frequency greater than 0.
- Double-check the total frequency by adding the frequency column, since every later calculation depends on it being correct.
Worked example
The table shows the number of pets owned by 20 students. Pets: 0 (frequency 5), 1 (frequency 7), 2 (frequency 6), 3 (frequency 2). Find the mean, median, mode and range.
- Total frequency = 5 + 7 + 6 + 2 = 20, which matches the 20 students.
- Frequency x value: 0x5=0, 1x7=7, 2x6=12, 3x2=6. Sum = 0+7+12+6 = 25.
- Mean = 25/20 = 1.25 pets.
- Mode: the highest frequency is 7, for the value 1, so the mode is 1 pet.
- Median: with 20 values, the median is the mean of the 10th and 11th values. Cumulative frequencies are 5, 12, 18, 20, so both the 10th and 11th values fall in the '1 pet' group (cumulative frequency reaches 12 there). Median = 1.
- Final answer: mean = 1.25, median = 1, mode = 1, range = 3 - 0 = 3.
Practice questions
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Q1A table shows number of siblings: 0 (freq 4), 1 (freq 10), 2 (freq 4), 3 (freq 2). Find the mode.Show answer
Answer: 1 (highest frequency, 10)
Q2Using the same table (0:4, 1:10, 2:4, 3:2), find the range.Show answer
Answer: 3 (3 - 0 = 3)
Q3Using the same table (0:4, 1:10, 2:4, 3:2), find the total number of students.Show answer
Answer: 20 (4+10+4+2 = 20)
Q4Using the same table (0:4, 1:10, 2:4, 3:2), find the mean number of siblings.Show answer
Answer: 1.2 (sum of f x x = 0+10+8+6 = 24, 24/20 = 1.2)
Q5Using the same table (0:4, 1:10, 2:4, 3:2), find the median number of siblings.Show answer
Answer: 1 (20 values; median is mean of 10th and 11th values; cumulative frequencies 4, 14, 18, 20 place both in the '1 sibling' group)
Q6A different table shows test scores out of 5 for 30 students: 1 (freq 3), 2 (freq 5), 3 (freq 9), 4 (freq 8), 5 (freq 5). Find the mean score.Show answer
Answer: 3.23 (2 dp) (sum of f x x = 3+10+27+32+25 = 97, 97/30 = 3.23)
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
The table shows the number of goals scored by a hockey team in 15 matches: 0 (freq 3), 1 (freq 6), 2 (freq 4), 3 (freq 2). Calculate the mean number of goals per match.
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The table shows the number of text messages sent by 25 students in one evening: 0 (freq 2), 1 (freq 5), 2 (freq 8), 3 (freq 6), 4 (freq 4). Find the median number of text messages.
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The table shows the number of DVDs owned by 12 families: 0 (freq 1), 1 (freq 3), 2 (freq 4), 3 (freq x), 4 (freq 1), where x is unknown and the total frequency is 12. Find x, then state the mode.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
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This topic is chapter 4 of GCSE Statistics Foundation Workbook 2 and chapter 8 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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