Averages and Measures of Spread

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infoWhy this? Grouped data requires us to reason from incomplete information, identifying a median class and estimating a mean when exact values are unavailable. This develops statistical judgement about approximation, distribution, and the limits of conclusions.

scheduleWhy now? Year 8 established averages from ungrouped frequency tables and cumulative counting. We now extend those methods to class intervals, midpoints, and cumulative frequencies, where answers must be interpreted as estimates rather than exact values.

Grouped data is organised into class intervals, and each class interval shows a range of values rather than individual data values.
The frequency of a class interval is the number of data values that fall within that interval.
The total frequency is the sum of all the frequencies in the table.
The median is the middle value of an ordered data set, so for grouped data we can only identify the class interval that contains the median, not the exact median value.
To find the median class interval, you first need the position of the median in the full data set.
For a data set with total frequency `n`, the median position is found using the middle of the data, commonly taken as `\frac{n}{2}` or `\frac{n+1}{2}`, depending on the method used.
Cumulative frequency is the running total of the frequencies as you move through the class intervals in order.
The median class interval is the first class interval whose cumulative frequency is greater than or equal to the median position.
A class midpoint is the value halfway between the lower and upper boundaries of a class interval.
The class midpoint is used to represent all the values in that class interval when estimating the mean.
The estimated mean from grouped data is calculated using `\frac{\sum(\text{midpoint} \times \text{frequency})}{\text{total frequency}}`.
The estimated mean is only an approximation because the exact data values within each class interval are not known.
Estimating the mean from grouped data assumes that the data values are spread evenly across each class interval.
If class intervals have different widths, the same method still works as long as the correct midpoint is used for each interval.
The estimated mean should usually be given to a sensible degree of accuracy that matches the context of the data.

rocket_launchYou must be able to

Add the frequencies to find the total frequency accurately.
Calculate cumulative frequencies in order by forming a running total for each class interval.
Find the median position from the total frequency and identify where this position falls in the cumulative frequency column.
State the class interval containing the median by choosing the first interval whose cumulative frequency reaches or passes the median position.
Calculate each class midpoint by finding the value halfway between the ends of the interval.
Multiply each class midpoint by its frequency to obtain the estimated total for that class.
Add the `\text{midpoint} \times \text{frequency}` values to find `\sum fx`.
Estimate the mean by dividing `\sum fx` by the total frequency and present the result clearly.
Explain that the answer is an estimate because grouped data does not show the exact original values.

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