Averages and Measures of Spread

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infoWhy this? Frequency tables summarise repeated values efficiently and allow measures of centre and spread to be calculated without reconstructing an entire list. This develops statistical fluency and reinforces the distinct meanings of mean, median, mode, and range.

scheduleWhy now? Building on Year 7 averages from raw lists, this unit introduces frequency as a compact representation and cumulative counting for positional measures. These methods are prerequisites for estimating averages and locating medians in grouped data.

A frequency table shows each data value and how many times it occurs, and the frequency is the number of occurrences.
The mode is the data value with the greatest frequency in the table.
If two or more data values share the highest frequency, the data set has more than one mode and there is no single modal value.
The range measures spread and is found by subtracting the smallest data value with a non-zero frequency from the largest data value with a non-zero frequency.
To find the median, the data must be considered in numerical order, which a frequency table already shows when the values are listed from smallest to largest.
The total number of data items is the sum of all the frequencies in the table.
When the total frequency is odd, the median is the single middle value in the ordered data set.
When the total frequency is even, the median is the mean of the two middle values in the ordered data set.
Cumulative frequency helps locate the median because it shows how the frequencies build up across the ordered values.
The mean is the sum of all the data values divided by the total number of data items.
For a frequency table, the total of the data values is found by multiplying each value by its frequency and then adding these products.
The mean from a frequency table can be written as `\text{mean} = \frac{\sum(\text{value} \times \text{frequency})}{\sum \text{frequency}}`.
A value with frequency zero does not appear in the data set, so it does not affect the mode, median, range, or mean.
The mode, median, and mean are different measures of central tendency, while the range is a measure of spread.

rocket_launchYou must be able to

Identify the mode from a frequency table by finding the value with the highest frequency.
Calculate the range from a frequency table by locating the smallest and largest values with non-zero frequencies and subtracting the smallest value from the largest value.
Find the total frequency by adding all the frequencies in the table accurately.
Locate the median from a frequency table by using the total frequency to find the middle position, then matching that position to the correct value in the ordered table.
Find the median when the total frequency is even by identifying the two middle positions and averaging the two corresponding values.
Calculate the mean from a frequency table by working out each `\text{value} \times \text{frequency}`, summing these products, and dividing by the total frequency.
Check a completed answer by interpreting it in context, for example confirming that the mode is an actual data value, that the median lies between the two middle values, and that the range is non-negative.

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