Averages from Tables (AQA GCSE Maths)

How do we find averages if there are lots of values?

• In reality there will be far more data to work with than just a few numbers
• In these cases the data is usually organised in such a way to make it easier to follow and understand
  • For example in a table or chart
• We can still find the mean, median and mode but have to ensure we understand what the table or chart is telling us

How do I find averages from a table or chart?

• Finding the median and mode from tables/charts is fairly straightforward once you understand what the table/chart is telling you
• Tables allow data to be summarised neatly
  • and quite importantly, they put the data into order

Finding the mean from (discrete) data presented in tables

• The mean can be found as you long as understand what a table is telling you
  • Tables tell us the data value
    • e.g. the number of pets per household
  • and the frequency of that data value
    • e.g. the number of households with that number of pets

   STEP 1
   Add a column to the table and work out "data value" × "frequency"
   (This is effectively doing the 'adding up' part of finding the mean in stages)

   STEP 2
   Find the total of the extra column to give the overall total of the data values

   STEP 3
   Find the mean by dividing this total by the total of the frequency column
   i.e.  divide the total of the data values by the number of data values

Finding the median from (discrete) data presented in tables

• The median is the middle value when the data is in order
• The position of the median can be found by using , where  is the number of data values
• Use the table to deduce where the  value lies
  • e.g. if the median is the 7^th value and the frequency of the first two rows are 4 and 7
    • the median will be one of the 7 values the second row of that table

Finding the mode (or modal value)

• The mode (or modal value) is simple to identify
  • Look for the highest frequency
    • and thus find the corresponding data value
  • Make sure you do not confuse the data value with the frequency!
    • The frequency (in a table) tells us the row the mode is in

What is grouped data and why use it?

• Some data for a particular scenario can vary a lot
  • For example the heights of people
    • particularly if you include a mixture of children and adults
• Data like height is also continuous (data that can be measured)
  • Such data is difficult, even using a table, to list every value
  • There is also little difference between someone who is, say, 176 cm tall and someone who is 177cm tall
• So we often group data into classes but that leads to one crucial point
  • When data is grouped, we lose the raw data
  • With height, we know that 10 people have a height between 150 cm and 160 cm
    • but we won't know exactly what those 10 heights are
• This means we cannot find the actual mean, median and mode from grouped data by their original definitions
  • but we can estimate the mean
    • look for the word estimate in questions - it's a big clue to use the method in this note!
  • we can talk about the class interval that the median lies in
  • and we can talk about the modal class (the class interval the mode lies in)

How do I find/estimate the mean from grouped data?

• There is one extra stage to this method compared to finding the mean from tables with discrete data
  • We use the class midpoints as our data values
  • For example, if heights are split into class intervals 150 ≤ x < 160, 160 ≤ x < 170, etc
    • the midpoints, and so data values, would be 155, 165, etc

STEP 1
Draw an extra two columns on the end of a table of the grouped data
In the first new column write down the midpoint of each class interval

STEP 2
Work out "frequency" × "midpoint" (This is often called fx )

STEP 3
Total the fx column, and if not already done nor mentioned in the question, total the frequency column to find the number of data values involved

STEP 4
Estimate the mean by using its formula; "total of fx" ÷ "no. of data values"

• Be careful with midpoints
  • not all class intervals will be of equal size
  • so there may not be a nice pattern to the midpoints

How do I find the class interval that the median lies in?

• Find the position of the median using , where  is the number of data values (total of the frequency column)
• Use the table to deduce the class interval containing the  value
  • e.g. if the median is the 7^th value and the frequency of the first two class intervals are 4 and 7
    • the median will lie in the second class interval of the table
• Note the language used when working with the median for grouped data
  • Rather than 'the median' we refer to the 'class interval containing the median'

How do I find the modal class interval?

• Similar to finding the median we are only interested in the class interval the modal value lies in
• Look for the highest frequency in the table
  • and then find the corresponding class interval
• Take care not to confuse the class interval with the frequency
  • The frequency tells us the class interval the modal value lies in