Averages & Range Notes

Cambridge IGCSE Maths: Revision notes

Key facts

  • Mean = total ÷ number of values. It uses every value but is pulled by outliers.
  • Median = middle value of the ordered data. It is not affected by outliers.
  • Mode = most common value. It works for non-numerical data too.
  • Range = highest − lowest; it measures spread, not average.
  • Frequency table mean =∑(value×frequency)∑frequency=\dfrac{\sum(\text{value}\times\text{frequency})}{\sum\text{frequency}}. Extended: IQR =Q3−Q1=Q_3-Q_1 and grouped means use midpoints.

Mean, median and mode

Each average describes the "typical" value differently, and suits different data.

The mean is the total divided by how many values there are. The median is the middle value of the ordered data (the mean of the two middle values if there is an even number). The mode is the most frequent value. Order the data before finding the median.

Mean

  • Add all, divide by how many
  • Affected by outliers

Median

  • Middle of ordered data
  • Not affected by outliers

Mode

  • Most frequent value
  • Works for categories

Worked example

Find the mean, median and mode of 4, 7, 3, 9, 4.

024681012MeanMedianAverageValue
  • Original data
  • With 40 added
Adding the outlier 40 to the data 3, 4, 4, 7, 9

What is the median of 9, 2, 6, 5?

Range

The range is the highest value minus the lowest, a measure of spread.

The range measures how spread out the data is. A small range means the values are close together; a large range means they are spread out. It is a measure of spread, not an average.

  • Rangehighest value − lowest value

Worked example

Find the range of 3, 4, 4, 7, 9.

What is the range of −3, 5 and 12?

Frequency tables

Multiply each value by its frequency for the mean; use cumulative frequency for the median.

The mode is the value with the highest frequency. For the median, find the position n+12\dfrac{n+1}2 and use cumulative frequencies to see which value covers it. For the mean, multiply each value by its frequency, add these and divide by the total frequency. At Core the values are individual, not grouped.

  • Mean∑(value×frequency)∑frequency\dfrac{\sum(\text{value}\times\text{frequency})}{\sum\text{frequency}}

Value 2

Frequency:
3
Value × frequency:
6

Value 3

Frequency:
5
Value × frequency:
15

Value 4

Frequency:
4
Value × frequency:
16

Value 5

Frequency:
2
Value × frequency:
10

Worked example

Find the mean, median and mode for the table (frequencies 3, 5, 4, 2 for values 2, 3, 4, 5).

Values 1, 2, 3 have frequencies 2, 5, 3. What is the mean?

Quartiles and grouped data

The IQR is Q3−Q1Q_3-Q_1; a grouped mean is an estimate using class midpoints.

In ordered data, the lower quartile Q1Q_1 is a quarter of the way through and the upper quartile Q3Q_3 is three-quarters of the way through. The interquartile range is Q3−Q1Q_3-Q_1: the spread of the middle 50%, less affected by outliers. For grouped data, estimate the mean with class midpoints, and the modal class is the class with the highest frequency.

  • IQRQ3−Q1Q_3-Q_1
  • Estimated mean∑(midpoint×frequency)∑frequency\dfrac{\sum(\text{midpoint}\times\text{frequency})}{\sum\text{frequency}}

Worked example

Find the IQR of 2, 4, 5, 7, 8, 10, 12, 15.

Worked example

Estimate the mean: classes 0≤x<100\le x<10, 10≤x<2010\le x<20, 20≤x<3020\le x<30 with frequencies 4, 6, 10 (midpoints 5, 15, 25).

Which is the modal class for frequencies 4, 6, 10 in the classes above?

Try an exam question

Here are six numbers: 6, 2, 9, 4, 4, 5. Work out (a) the mean, (b) the median, (c) the range.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.