Averages and Measures of Spread

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infoWhy this? Measures of centre and spread allow us to summarise a data set and compare its typical values and variability. Understanding the distinct meanings of mean, median, mode, and range supports critical interpretation rather than mechanical calculation.

scheduleWhy now? This unit follows earlier work on organising and reading data, so we can now summarise the values those representations contain. Secure methods with raw data prepare us to calculate averages from frequency tables and estimate from grouped data.

The mean, median, and mode are all measures of average, while the range measures how spread out the data is.
The mode is the value that occurs most often in a data set.
A data set can have more than one mode if two or more values share the highest frequency.
A data set has no mode if every value occurs the same number of times.
The median is the middle value in a data set after the values have been put in order.
To find the median correctly, the data must be arranged from smallest to largest or largest to smallest.
If a data set has an odd number of values, the median is the single value in the middle position.
If a data set has an even number of values, the median is the mean of the two middle values.
The mean is found using `\text{mean} = \text{total of all values} \div \text{number of values}`.
The range is found using `\text{range} = \text{largest value} - \text{smallest value}`.
If all values in a data set are equal, the range is 0.
Repeated values must still be counted separately when finding the mean, median, and mode.
The mean does not have to be one of the values that appears in the data set.
The median is determined by the middle position or positions after all values, including repeats, have been ordered.
The mode depends on frequency, not on the size of the value.

rocket_launchYou must be able to

Identify the mode by counting how often each value appears and selecting the value or values with the greatest frequency.
Order a data set from smallest to largest before finding the median, keeping repeated values in the list.
Decide whether a data set has an odd or even number of values before choosing the method for the median.
Find the median of a data set with an odd number of values by locating the single middle value in the ordered list.
Find the median of a data set with an even number of values by locating the two middle values and calculating their mean.
Calculate the mean by adding all the values accurately and dividing by the number of values in the data set.
Calculate the range by identifying the largest and smallest values and subtracting the smallest from the largest.
Check a completed answer by matching the method to the measure, for example using frequency for mode, position for median, total divided by number for mean, and subtraction for range.

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