4.1.3 Statistical Measures

What are the mean, mode and median?

• Mean, median and mode are measures of central tendency
  • They describe where the centre of the data is
• They are all types of averages
• In statistics it is important to be specific about which average you are referring to
• The units for the mean, mode and median are the same as the units for the data

How are the mean, mode, and median calculated for ungrouped data?

• The mode is the value that occurs most often in a data set
  • It is possible for there to be more than one mode
  • It is possible for there to be no mode
    • In this case do not say the mode is zero
• The median is the middle value when the data is in order of size
  • If there are two values in the middle then the median is the midpoint of the two values
• The mean is the sum of all the values divided by the number of values

• • Where  is the sum of the n pieces of data
  • The mean can be represented by the symbol μ
• Your GDC can calculate these statistical measures if you input the data using the statistics mode

Worked Example

What are quartiles?

• Quartiles are measures of location
• Quartiles divide a population or data set into four equal sections
  • The lower quartile, Q₁ splits the lowest 25% from the highest 75%
  • The median, Q₂ splits the lowest 50% from the highest 50%
  • The upper quartile, Q₃ splits the lowest 75% from the highest 25%
• There are different methods for finding quartiles
  • Values obtained by hand and using technology may differ
• You will be expected to use your GDC to calculate the quartiles

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

• • • This is given in the formula booklet
• The units for the range and interquartile range are the same as the units for the data

Worked Example

What are the standard deviation and variance?

• The standard deviation and variance are both measures of dispersion
  • They describe how spread out the data is in relation to the mean
• The variance is the mean of the squares of the differences between the values and the mean
  • Variance is denoted σ²
• The standard deviation is the square-root of the variance
  • Standard deviation is denoted σ
• The units for the standard deviation are the same as the units for the data
• The units for the variance are the square of the units for the data

How are the standard deviation and variance calculated for ungrouped data?

• In the exam you will be expected to use the statistics function on your GDC to calculate the standard deviation and the variance
• Calculating the standard deviation and the variance by hand may deepen your understanding
• The formula for variance is 
  • This can be rewritten as

• The formula for standard deviation is 
  • This can be rewritten as

• You do not need to learn these formulae as you will use your GDC to calculate these

Worked Example