Representing Data Numerically (AQA Level 3 Mathematical Studies (Core Maths): Paper 1: Data, Finance, Estimation & Modelling): Revision Note

Mean, Median & Mode

What is the mode?

• The mode is the value that appears the most often

  • The mode of 1, 2, 2, 5, 6 is 2

• There can be more than one mode

  • The modes of 1, 2, 2, 5, 5, 6 are 2 and 5

• The mode can also be called the modal value

• In some situations there may be no mode

What is the median?

• The median is the middle value when you put values in size order

  • The median of 4, 2, 3 can be found by

    • ordering the numbers: 2, 3, 4

    • and choosing the middle value, 3

• If you have an even number of values, find the midpoint of the middle two values 

  • The midpoint is the sum of the two middle values divided by 2

  • The median of 1, 2, 3, 4 is 2.5

    • 2.5 is the midpoint of 2 and 3

What is the mean?

• The mean is the sum of the values divided by the number of values

  • The notation, $\overline{x}$, is used to represent the mean

  • The mean of 1, 2, 6 is (1 + 2 + 6) ÷ 3 = 3

• The mean can be fraction or a decimal

  • It may need rounding

  • You do not need to force it to be a whole number

    • You can have a mean of 7.5 people, for example!

• The mean is often selected because it uses all of the data

How do I calculate the mean from a frequency table?

• To find the mean from a frequency table of ungrouped data, use the formula

  $\overline{x}=\frac{\sum _{}fx}{\sum _{}f}$

  • where $\sum _{}fx$ is the sum of each data item, $x$, multiplied by its corresponding frequency, $f$

  • and $\sum _{}f$ is the sum of all of the frequencies

• To find the mean from a frequency table of grouped data

  • Use the same formula as for a frequency table

  • Use the mid-interval value (midpoint) of each group as the value for $x$

How do I know which average to use?

• The mode, median and mean are different ways to measure an average

  • Units for the mean, median and mode are the same as for the data set

• In certain situations it is better to use one average over another

• For example:

  • If the data has extreme values (outliers)

    • Don't use the mean (it's badly affected by extreme values)

  • If the data has more than one mode 

    • Don't use the mode as it is not clear

  • If the data is non-numerical, like dog, cat, cat, fish

    • You can only use the mode

Range, Quartiles & Outliers

What are quartiles?

• Quartiles are measures of location

• Quartiles divide a population or data set into four equal sections

  • The lower quartile, $Q_1$, splits the lowest 25% from the highest 75%

  • The median, $Q_2$, splits the lowest 50% from the highest 50%

  • The upper quartile, $Q_3$, splits the lowest 75% from the highest 25%

• There are different methods for finding quartiles, depending on the number of items in the data set, $n$

  • First, list the items in size order

  • When finding the median and quartiles from raw data:

    • The median will be at position $\frac{n+1}{2}$

    • The lower quartile will be at position $\frac{n+1}{4}$

    • The upper quartile will be at position $\frac{3\left(n+1\right)}{4}$

  • For larger data sets:

    • The median will be at position $\frac{n}{2}$

    • The lower quartile will be at position $\frac{n}{4}$

    • The upper quartile will be at position $\frac{3n}{4}$

    • The use of $n+1$ rather than $n$ will still be accepted however

What are the range and interquartile range?

• The range and interquartile range are both measures of dispersion

  • They describe how spread out the data is

• The range is the largest value of the data minus the smallest value of the data

• The interquartile range is the range of the central 50% of data

  • It is the upper quartile minus the lower quartile

$\mathrm{IQR}=Q_3-Q_1$

• The units for the range and interquartile range are the same as the units for the data

• The range can be affected by outliers (extreme values)

  • Outliers will not affect the interquartile range

What are outliers?

• Outliers are extreme data values that do not fit with the rest of the data

  • They are either a lot bigger or a lot smaller than the rest of the data

• Outliers are defined as values that are more than $1.5\times \mathrm{IQR}$ from the nearest quartile

  • $x$ is an outlier if $x<Q_1-1.5\times \mathrm{IQR}$ or $x>Q_3+1.5\times \mathrm{IQR}$

• Outliers can have a big effect on some statistical measures

Should I remove outliers?

• The decision to remove outliers will depend on the context

• Outliers should be removed if they are found to be errors

  • The data may have been recorded incorrectly

  • For example, the number 17 may have been recorded as 71 by mistake

• Outliers should not be removed if they are a valid part of the sample

  • The data may need to be checked to verify that it is not an error

  • For example, the annual salaries of employees of a business might appear to have an outlier but this could be the director’s salary

Standard Deviation

What is standard deviation?

• The standard deviation, $\sigma$, is a measure of dispersion

  • It describes how spread out the data is in relation to the mean

  • If greater the value of the standard deviation, the more spread out the data is

• The units for the standard deviation are the same as the units for the data

How is standard deviation calculated?

• The standard deviation is the square root of the mean of the squares of the differences between the values and the mean

• The formula used in this course for standard deviation is the standard deviation for a sample

${\sigma }_{n-1}=\sqrt{\frac{{\displaystyle \sum {\left(x-\overline{x}\right)}^2}}{n-1}}$

• You can calculate the standard deviation of a small data set by hand

• You can also enter the data to your calculator and use the stats calculation options to calculate the standard deviation of a data set