Mode, Median & Arithmetic Mean (Edexcel GCSE Statistics)

Mode, Median & Mean from Discrete Data

Why do we have different types of average?

• You’ll hear the phrase “on average” used a lot

  • For example

    • by politicians talking about the economy

    • by sports analysts

• However not all data is numerical

  • e.g. the party people voted for in the last election

• And even when data is numerical

  • some of the data may lead to misleading results

• This is why we have 3 types of average

What are the three types of average?

1. Mean

• This is what people usually mean when they say “average”

  • In an ideal world where everybody had the same amount of some resource

  • the mean is the amount of that resource that each person would have

• It is also known as the arithmetic mean

• It is the total of all the values divided by the number of values

  • 

    • i.e. add up all the data values

    • then divide by how many values there are

• Problems with the mean occur when there are one or two unusually high (or low) values in the data (outliers)

  • These can make the mean too high (or too low) to accurately represent any patterns in the data 

2. Median

• This is similar to the word medium, which can mean 'in the middle'

• So the median is the middle value

  • But beware, the data has to be arranged into numerical order first!

• Use the median instead of the mean if you don't want extreme values (outliers) affecting the average

• If there are an odd number of values, there will only be one middle value

  • This will be the ^th value

    • e.g. for 35 data values, and

    • So the median will be the 18^th value

• If there are an even number of values there will be two values in the middle

  • In this case we take the halfway point between these two values

  • Often the halfway point is obvious

  • If not, add the two middle values and divide by 2

    • this is the same as finding the mean of the two middle values

3. Mode

• Think of MOde as meaning the Most Often

  • i.e. it is the value that occurs the greatest number of times

• It is often used for things like “favourite …” or “… sold the most” or “… were the most popular”

• Not all data is numerical and that is where the mode is especially useful

  • But be aware that the mode can be applied to numerical data

    • e.g. data about sales of clothing or shoes in different sizes

    • mode would be the best average for determining demand for the sizes

• The mode is sometimes referred to using the word modal

  • e.g. you may see a phrase like “modal value”

  • This means the same thing, the value occurring most often

• Sometimes no value occurs more often than any other

  • In this case we say there is no mode

• If two values occur most often we may say there are two modes

  • or say that the data set is bi-modal

  • Whether it is appropriate to do this will depend on what the data is about

How do changes in the data affect the average?

• You should be able to determine how a change in the data can affect the mean, median or mode

  • For example adding or removing data values to the set

• You can always recalculate the averages using the changed data set

  • This may be necessary if you need the exact values of the new averages

• But sometimes you can use logic to decide what kind of change will occur

• For the mode

  • If an added data value is equal to the modal value, it will not change the mode

  • If a removed data value is not equal to the modal value, it will not change the mode

  • Otherwise recalculate

• For the mean

  • If an added data value is

    • greater than the mean, the mean will increase

    • less than the mean, the mean will decrease

  • If a removed data value is

    • greater than the mean, the mean will decrease

    • less than the mean, the mean will increase

  • (If an added or removed data value is equal to the mean, the mean will not change!)

  • Recalculate to find the exact value of a changed mean

• For the median

  • You will need to examine the changed data set to decide if the median has changed

  • Make sure the values in the changed set are written in order!

Problem-solving with the Mean

What does problem-solving with the mean involve?

• The mean is calculated from a formula

  • You could be asked questions that require

    • using the formula 'backwards'

    • rearranging the formula

• 

  • This is a formula involving 3 quantities

  • If you know any 2, you can find the other one

    • 

    • 

What types of problems might I need to solve?

• Typical questions ask you to either

  • work backwards from a known mean or

  • combine means for two data sets

• An exam question can always ask something unusual that you haven’t seen before

  • So you can't just practice 'every type of question' for this topic

• You will need to make sure you understand

  • what the mean is

  • how it works

  • and what it shows

How do I solve problems involving the mean?

• Known mean, unknown data value

  • This is working backwards from the mean, to an unknown data value

  • Call the unknown data value , say

  • Using the 'formula' for the mean, set up an equation in 

  • Rearrange and solve the equation to find , the unknown data value

• Combined means for two data sets

  • This is where we know the mean for two different data sets but would like to know the overall mean

  • We need to find the overall total of values from both data sets

    • then divide by the total number of values across both data sets 

  • Alternatively we may know the overall mean and want to

    • work back to the mean of one or both of the data sets

    • or to an unknown data value

• Others

  • Due to the problem solving nature of such questions there will be variation in question styles

  • The above two should give you a good idea and cover the vast majority of questions

  • The best way to start tackling questions with the mean is to

    • write down the quantities you do know

    • write down those you don't know

    • use the 'formula' for the mean to link the unknown and known values

Mode, Median & Mean from Tables & Charts

How can I find averages if there are lots of values?

• In the real world there will usually be more data to deal with than just a few numbers

• The data can be organised in a way that makes it easier understand

  • For example in a table or chart

• We can still find the mean, median and mode

  • But we have to understand what the table or chart is telling us

• Be careful - we are not talking here about tables for grouped data

  • In a grouped data table, the individual data values are no longer available

  • See the 'Mode & Mean from Grouped Data' and 'Linear Interpolation' revision notes

• The tables discussed in this note give the frequencies for each data value

  • That means the entire original data set is still available

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 usefully!) they put the data into order

• The instructions here show how to find averages from tables

  • For charts it is often easiest to turn the chart into a table first

    • See the Worked Example

  • But you can use the same methods directly from a chart if you feel confident doing so

Finding the mean from (discrete) data presented in tables

• Tables tell us

  • the data value

    • e.g. the number of pets per household

  • and the frequency of that data value

    • i.e., how many times that data value occurred

    • 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 doing the 'adding up' part of finding the mean

  • We're just doing it one data value at a time

• 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

  • e.g. if , then

    • The median is the 18^th value

  • Or if , then

    • The median is midway between the 24^th and 25^th 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 9

    • then the median will be one of the 9 values in 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

    • i.e. the data value that occurs the most times

  • The corresponding data value is the mode

  • Make sure you do not confuse the data value with the frequency!