Frequency Tables (CIE A Level Maths: Probability & Statistics 1)

In most cases in your statistics course, you will come across data that is presented in a frequency table. These allow data to be summarised and make them easier to work with.

A frequency table for ungrouped data shows the frequency of individual data values

Why use a frequency table for ungrouped data?

• When collecting large amounts of raw data, it is quicker to use a tally system and then collate the results into a frequency table
• Ungrouped frequency tables are normally used for numerical, discrete data
• Organising data into a frequency table makes it easier to work with
  • Calculating averages, ranges and summary statistics can be done much quicker from a frequency table than from raw data
• It gives a clear pattern of the data
  • It is easy to quickly see where most of the data are and to see extreme values
• A frequency table for ungrouped data keeps all of the original data values
  • It is still possible to calculate the actual averages, ranges and summary statistics

How are mean, median and mode calculated from an ungrouped frequency table?

• 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 an ungrouped frequency table the data value with the highest frequency will be the mode
• The median is the middle value when the data is in order of size
  • To find the median from an ungrouped frequency table, add the frequencies together until you reach the value that is half of the total
  • For a data set of values, calculate  and the median will be whichever data value corresponds with this frequency
• The mean is the sum of all the values divided by the number of values in the data set
  • To find the mean, multiply each data value by its corresponding frequency, find the sum of these values and then divide by the total frequency
  • The notation for the sum of the values is 
  • The formula for the mean from a frequency table is 

A frequency table for grouped data is usually used for large amounts of continuous data. They shows the frequency of data values that are within a particular group or class.

What are the advantages and disadvantages of grouping data?

• Grouping data is especially useful when data can take a large range of different values
• Trends and patterns can be easily spotted when data has been grouped
• Calculations are much quicker with data that has been grouped
• It is important to be aware, however, that grouped frequency tables also have some negatives
  • The actual data values are lost when data is grouped
  • It is only possible to calculate estimated averages, ranges and summary statistics 

Notation for grouped frequency tables

• When grouping data, it is important to be clear about which group or class any data value should be entered into
  • A group entry of 10 – 20 followed by 20 – 30 would be unclear because the number 20 could be entered into both groups
    • If the data are discrete, this could be written as 10 – 19 and 20 – 29
    • For continuous data, this could be changed to 10 – and 20 –
    • This would most likely be represented as  and 
• Most commonly inequalities are used to group continuous data as they leave no ambiguity
• If the data are continuous, always check that there are no gaps between upper boundary of a class and the lower boundary of the next class
  • If there are gaps you will need to close these gaps by changing the boundaries before carrying out any calculations
    • For example, the groups followed by will become and 
    • Check the inequality signs carefully
• Be careful when deciding what category the data falls into, taking the group for example,
  • if the data had been rounded it would take the form  as described above
  • if the data had been truncated however, then the boundaries would become 
    • this is most likely to happen with age, which is technically a continuous variable due to being able to take any, however we would usually consider age by counting years

Finding averages from grouped frequency tables

• Instead of finding the mode, when working with a grouped frequency table we instead find the modal class
  • This will be the class (group) with the greatest frequency
• It is only possible to calculate an estimate for the mean and the median from a grouped frequency table
• Calculating the estimated mean is the same as for ungrouped frequency tables, however you will need to find the midpoints first
  • the midpoint is the mean of the upper and lower values in the class boundaries
  • multiply the midpoint by its corresponding frequency, find the sum of these values and then divide by 
• Calculating the estimated median is more complicated and questions will only ask for the class that the estimated median would be in