2.1.1 Basic Statistical Measures

What are mean, median and mode?

• Mean, median and mode are measures of location
  • A measure of location gives information about where data is in the number system
  • 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

How are mean, median and mode calculated?

• You should already be familiar with finding the mean, median and mode from raw, ungrouped data
• The mode is the value that occurs most often in a data set
  • In a frequency table the group or class that occurs most often will be referred to as the modal class
  • A data set with more than one mode is bimodal
• The median is the middle value when the data is in order of size
  • If there are two values in the middle of the data set, the median is the midpoint of the two values
  • If finding median from a frequency table find the cumulative frequency first and find the group or class where the middle value will lie
  • You may have to use linear interpolation when finding the median from a grouped frequency table
• The mean is the sum of all the values divided by the number of values in the data set

What are summary statistics and their notation?

• Summary statistics are information that summarises a set of data values
• For  items in a data set:
  • The sum of the data is represented by

• • • This is usually written  and reads as ‘sigma x’

• • The mean of the data is represented by

• • • This reads as ‘x bar’
• You will come across more summary statistics later in the course

How do we choose the best measure of central tendency?

• It is often better to use one of the averages over the others, depending on the data set
  • It’s a good idea to be aware of the advantages and disadvantages of the use of each average
• The mean uses all of the data values, this is good for a large data set where all of the values are close together, but also means that the mean can be affected by extreme values
• The median is not affected by very high or low values so is a good average to use in data sets with extreme values
• The mode is very useful in a lot of practical situations, however often there may be more than one mode, no mode or even a mode that is nowhere near the middle of the data set

Worked Example

What are quartiles and percentiles?

• Quartiles and percentiles 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, is the value that is 50% of the way through the data
  • The upper quartile, Q₃, splits the lowest 75% from the highest 25%
• Percentiles divide the data into 100 parts
  • The 70th percentile lies seven-tenths of the way through the data
    • 70% of the data is below it and 30% is above it

How are quartiles calculated?

• For a data set of size, ,
  • To find the lower quartile, calculate 
    • If is an integer then the lower quartile is the midpoint of the corresponding value and the one above it
    • If is not an integer then the lower quartile is the value corresponding to the next integer up
  • To find the upper quartile, calculate 
    • If is an integer then the upper quartile is the midpoint of the corresponding value and the one above it
    • If is not an integer then the upper quartile is the value corresponding to the next integer up
• You can also use your calculator to find the quartiles, make sure you know how to put your calculator into STAT mode, enter the data and find the values of Q₁, Q₂ and Q₃
  • You could be expected to be able to do this in the exam

What are the range and interquartile range?

• The range and interquartile range are both measures of spread
• A measure of spread gives information about how spread out the data set is
• The range is the difference between the largest and smallest values in the data set
  • All data points in the set will be included in the range, including extreme values
• The interquartile range is the difference between the upper quartile and the lower quartile
  • Only the middle 50% of the data is included in the interquartile range
  • It is not affected by extreme values
  • Sometimes an interpercentile range could be asked for, this is the difference between two given percentiles
    • For example, the 20th to 80th interpercentile range would be the difference between the 80th percentile and the 20th percentile
• The units for range and interquartile range are the same as the units for the original data

Worked Example

Exam Tip