Frequency Tables
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.
Ungrouped Data
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
Worked Example
The frequency table below gives information about the shoe size of a group of year 12 students.
Shoe Size  37  37.5  38  38.5  39  39.5  40  40.5  41 
Frequency  3  3  12  5  16  9  7  4  1 
Exam Tip
 Use common sense when checking your answers, is your mean within the range of the data? Does is seem right? A mean of 140 for example could not be correct if the data is about ages of students taking an exam at university.
Grouped Data
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
 A group entry of 10 – 20 followed by 20 – 30 would be unclear because the number 20 could be entered into both groups
 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
 If there are gaps you will need to close these gaps by changing the boundaries before carrying out any calculations
 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 can involve using linear interpolation
Linear interpolation from grouped data
 For grouped data, the median, quartiles and percentiles can be found using a process called linear interpolation
 Interpolation uses the assumption that the data values are evenly spread throughout each class
 Once you know which data value you are looking for, interpolation uses proportion for find how far through each class the data value should be
 It is important to remember to add on the lowerclass boundary after finding the correct data value
 Follow these steps to use linear interpolation to find the median, quartiles or percentiles:
Step 1. Find the place within the data where the value you are looking to find is positioned, e.g. the median will be the th value
Step 2. Add up the frequencies until you know which class this value will fall into (this is finding the cumulative frequency)
Step 3. Write the upper and lowerclass boundaries of this group on the upper side of a simple number line
Step 4. On the other side of the same number line add the lower and upper values of the cumulative frequency for this group to the ends of the line
Step 5. Use proportion to calculate the data value that is the same fraction of the way through the group on the upper side of the number line as where the value you are looking for would sit on the lower side of the number line


 To see these steps in action, go to part (iv) of the worked example

 This method can be used to find any percentile, for example the 81^{st} percentile will be the value that is ths of the way through the data
Worked Example
The table below shows the heights in cm of a group of year 12 students.
Height, 
Frequency 
3 

5 

9 

7 

1 
Exam Tip
 There can be a lot of calculations when working with grouped frequency tables, be extra careful when using your calculators as it is easy to make small errors with these questions. Use the table and add information to the table as you go as there will be marks available for showing work within the methods. Make sure you know how to use your calculator to find all of the summary statistics as this can save time and reduce errors.