Averages from Grouped Data

Averages from Grouped Datavideo

Averages from grouped data

What is grouped data?

• Data can be collected into groups or class intervals

  • It is useful for organising data if you have a lot of individual data points

  • You can present grouped data in a grouped frequency table

• Grouped data may be discrete or continuous

  • Discrete data is numerical data that can only take on specific values, it needs to be counted

    • E.g. Shoe size

  • Continuous data can take any value within a range of infinite values, it needs to be measured

    • E.g. Length of a foot in cm

Why can I only find an estimate for the mean from grouped data?

• It is impossible to find the mean for grouped data, because we don't have access to the original data values

  • i.e. there is no way to find the exact sum of all the data values

  • so we can't use the formula, $\mathrm{mean}=\frac{\mathrm{sum} \mathrm{of} \mathrm{values}}{\mathrm{number} \mathrm{of} \mathrm{values}}$

• However we can estimate the mean for grouped data

  • To do this we use the class midpoints as our data values

    • e.g. if a class interval is 150 ≤ x < 160

    • we assume that all the data values are equal to the midpoint, 155

How do I find an estimate for the mean from grouped data?

• To find an estimate for the mean from grouped data, complete the following steps:

• 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

  • If the midpoint isn't obvious, add the endpoints and divide by 2

    • e.g. if a class interval is 150 ≤ x < 160

    • the midpoint is $\frac{150+160}{2}=\frac{310}{2}=155$

• STEP 2
  Calculate "frequency" × "midpoint" (this is often called fx)

  • Write these values in the second column you added to the table

• STEP 3
  Find the total for the fx column

  • If the question does not tell you the total number of data values (i.e. the total frequency), find the total of the frequency column also

• STEP 4
  Estimate the mean by using the formula

  • $\mathrm{estimated} \mathrm{mean}=\frac{\mathrm{total} \mathrm{of} \left(\mathrm{midpoints}\times \mathrm{frequencies}\right) }{\mathrm{total} \mathrm{frequency}}$

  • i.e. divide the total of the fx column by the total number of data values

How do I find the modal class?

• For grouped data we talk about the modal class instead of the mode

  • This is the class with the highest frequency

• Find the highest frequency in the table

  • The corresponding class interval tells you the modal class

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

• Find the position of the median using $\frac{n+1}{2}$, where $n$ is the number of data values (total of the frequency column)

• Use the table to deduce the class interval containing the ${\left(\frac{n+1}{2}\right)}^{\mathrm{th}}$ 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 that rather than 'the median' we refer to the 'class interval containing the median'