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Statistics

Statistics

Averages and range

Mode Median Mean- Raw Data Core + Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. Averages of a list
  4. Averages from a frequency table
  5. Stem-and-leaf diagrams
  6. Quartiles and interquartile range Extended
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 9.1-9.3: averages and spread
  • 9.3Find the mean, median, mode and range of a list or a frequency table.
  • 9.3Choose the best average for a purpose, and work backwards from a mean.
  • 9.4Read and draw a stem-and-leaf diagram.
  • 9.3Find quartiles and the interquartile range of a list Extended.

Before you start: ordering numbers and division. Next: Grouped data.

Formula bank

MeanLearn it\(\text{mean}=\dfrac{\text{total of the values}}{\text{number of values}}\)Use it when 'mean', 'average' with a calculation.
MedianLearn itthe middle value in order: position \(\tfrac{n+1}{2}\)two middle values? halfway between them.Use it when 'median'.
ModeLearn itthe most common valuecan be more than one, or none.Use it when 'mode', 'most common'.
RangeLearn itlargest − smallesta measure of SPREAD.Use it when 'range', 'how spread out'.
Frequency table meanLearn it\(\dfrac{\sum fx}{\sum f}\)x = value, f = how many times.Use it when a frequency table.
QuartilesLearn itlower quartile at \(\tfrac{n+1}{4}\), upper at \(\tfrac{3(n+1)}{4}\); IQR = UQ − LQthe middle half of the data.Use it when 'interquartile range'.

Averages of a list

An average is one number that represents the data. Data: 3, 7, 4, 9, 7, 2, 8. In order: 2, 3, 4, 7, 7, 8, 9. Mean \(=\tfrac{40}{7}=5.71\); median = 7 (the 4th of 7); mode = 7; range = 9 − 2 = 7.

General methodHow to find each average

  1. Mean: add everything, divide by how many there are.
  2. Median: put the values in ORDER first, then take the middle one. Even number of values? Halfway between the two middle ones.
  3. Mode: the value that appears most.
  4. Range: largest − smallest (one number, not '2 to 9').

Example Find the median and mean of 12, 15, 9, 20, 14, 10.

Six values: the median is halfway between the 3rd and 4th.

  1. in order: 9, 10, 12, 14, 15, 20
  2. median \(=\tfrac{12+14}{2}=13\)
  3. mean \(=\tfrac{80}{6}=13.33\)

Answer median 13, mean 13.33

Example The mean of five numbers is 8. Four of them are 6, 9, 5 and 10. Find the fifth.

Total = mean × how many.

  1. total \(=5\times8=40\)
  2. \(40-(6+9+5+10)=10\)

Answer 10

Which average? The mean uses every value but is pulled by one very big or small value (an outlier). The median is not affected by outliers. The mode is the only one for non-number data (favourite colour).

Try it yourself

1
Range of 12, 15, 9, 20, 14, 10?
Show answer

Answer 11

2
Mean of 3, 7, 4, 9, 7, 2, 8 (2 d.p.)?
Show answer

Answer 5.71

Averages from a frequency table

Goals x01234
Frequency f47531
f × x071094

General methodHow to find averages from a frequency table

  1. Mean: add a row f × x; mean \(=\sum fx\div\sum f\) (divide by the number of matches, not by 5).
  2. Median: find the position \(\tfrac{n+1}{2}\), then count along the frequencies to it.
  3. Mode: the value with the highest frequency (not the frequency itself).

Example Find the mean, median and mode of the goals table.

20 matches.

  1. mean \(=\tfrac{30}{20}=1.5\)
  2. median: 10th and 11th values; first 4 are 0, next 7 are 1, so both are 1
  3. mode: 1 (frequency 7)

Answer mean 1.5, median 1, mode 1

Try it yourself

1
Shoe sizes 5, 6, 7, 8, 9 with frequencies 2, 5, 8, 4, 1. Mean (2 d.p.)?
Show answer

Answer 6.85

Stem-and-leaf diagrams

StemLeaf
21 3 3
30 4 5 8
41 2 7
52

Key: 2 | 1 means 21. The diagram keeps every value and shows the shape. Here: 11 values, median = 6th = 35, mode 23, range 52 − 21 = 31. Leaves must be in order, and there must be a key.

Quartiles and interquartile range Extended

The quartiles split ordered data into four quarters. For 2, 3, 4, 7, 7, 8, 9 (n = 7): lower quartile = \(\tfrac{8}{4}\) = 2nd value = 3; upper quartile = 6th value = 8; IQR = 8 − 3 = 5. The IQR measures the spread of the middle half, so it ignores outliers.

Try it yourself

1
For the stem-and-leaf data (11 values), find the IQR.
Hint
3rd value and 9th value.
Show answer

Answer 19

Ways they test it

Type 1 Average of a list

"Find the median." (1-2 marks)

Spot it a list of values.

  1. Order first.
  2. Middle position.

Type 2 Frequency table mean

"Calculate the mean number of goals." (3 marks)

Spot it a frequency table.

  1. \(\sum fx\div\sum f\).

For example goals table

  1. \(30\div20\)

Answer 1.5

Type 3 Missing value

"The mean is 8. Find the missing number." (2 marks)

Spot it a mean and a gap.

  1. Total = mean × n.
  2. Subtract the known values.

Type 4 Choose and explain

"Which average best represents…? Give a reason." (1 mark)

Spot it 'which average'.

  1. Outlier? median.
  2. Non-number data? mode.

Type 5 Stem-and-leaf

"Find the median and the interquartile range." (3 marks)

Spot it a stem-and-leaf diagram.

  1. Count the leaves.
  2. Positions, then read the values with the key.

Mistakes to avoid

  • ✗ Median without ordering✓ order the data first
  • ✗ Table mean ÷ number of rows✓ ÷ \(\sum f\)Divide by the number of data items.
  • ✗ Mode = the highest frequency (7)✓ the VALUE with that frequency (1)
  • ✗ Range written as '2 to 9'✓ one number: 7

Summary

  • Mean = total ÷ count; median = middle in order; mode = most common; range = biggest − smallest.
  • Frequency table: mean \(=\sum fx\div\sum f\).
  • Working back: total = mean × count.
  • IQR = UQ − LQ (middle half).

If you see …

  • an outliermedian is better than mean.
  • a frequency tableadd an f × x row.

Practice

Practice: easy, medium, hard

Easy

1
Median of 3, 7, 4, 9, 7, 2, 8?
Show answer

Answer 7

2
Mode of 3, 7, 4, 9, 7, 2, 8?
Show answer

Answer 7

3
Median of 12, 15, 9, 20, 14, 10?
Show answer

Answer 13

Medium

1
Goals table (0:4, 1:7, 2:5, 3:3, 4:1). Mean?
Guide me

What is it asking? a mean

What do we know? \(\sum fx=30\)

Plan \(\div20\)

Show working
  1. 1.5

Answer 1.5

2
Mean of five numbers is 8; four are 6, 9, 5, 10. Fifth?
Guide me

What is it asking? a value

What do we know? total 40

Plan 40 − 30

Show working
  1. 10

Answer 10

3
Goals table: median?
Guide me

What is it asking? a median

What do we know? 10th and 11th values

Plan count along: 4 zeros, then 7 ones

Show working
  1. 1

Answer 1

Hard

1
Shoe sizes 5-9 with frequencies 2, 5, 8, 4, 1. Mean (2 d.p.)?
Guide me

What is it asking? a mean

What do we know? \(\sum fx=137\)

Plan \(\div20\)

Show working
  1. 6.85

Answer 6.85

PartAnswer / workingMarksGuidance
\(\sum fx=137\)M1
\(\div20\)M1
6.85A1
Total: 3 marks
2
Mean of 12, 15, 9, 20, 14, 10 (2 d.p.)?
Guide me

What is it asking? a mean

What do we know? total 80

Plan \(\div6\)

Show working
  1. 13.33

Answer 13.33

PartAnswer / workingMarksGuidance
80/6M1
13.33A1
Total: 2 marks
3
Shoe sizes table: median?
Guide me

What is it asking? a median

What do we know? 10th and 11th values

Plan 2 + 5 = 7, so both are in the 8 sevens

Show working
  1. 7

Answer 7

PartAnswer / workingMarksGuidance
7B1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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