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Measures of Centre and Spread
For the data set 3, 7, 8, 8, 10, 12, 15, which measure of centre best represents the data? Why?
Learning Intentions
Know
- Mean
- Median
- Mode
- Range
- Interquartile range
Understand
- When median is preferable to mean
- How outliers affect each measure
Can Do
- Calculate mean, median, mode
- Find range and IQR
- Choose appropriate measures for given data
Mean
The mean is the sum of all values divided by the number of values.
$ar{x} = dfrac{sum x}{n}$
The mean uses all data points but is affected by outliers.
Example: 2, 4, 6, 8, 100. Mean = 24, but most values are much smaller.
Median and Mode
The median is the middle value when data is ordered.
For $n$ values: position = $(n+1)/2$
If $n$ is even, average the two middle values.
The mode is the most frequently occurring value.
Example: 3, 5, 7, 8, 8, 10. Median = $(7+8)/2 = 7.5$. Mode = 8.
Range and IQR
Range = maximum − minimum
Interquartile Range (IQR) = Q3 − Q1
Where Q1 is the median of the lower half and Q3 is the median of the upper half.
The IQR measures the spread of the middle 50% of data and is not affected by outliers.
Outliers are often defined as values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.
Check Understanding
For data 2, 5, 7, 8, 10, 12, 20: find mean, median, mode, range, and IQR. Identify any outliers.
Measures of Centre and Spread
Find mean, median, mode for: 4, 6, 6, 8, 10, 12.
Mean = $(4+6+6+8+10+12)/6 = 46/6 approx 7.67$
Median = $(6+8)/2 = 7$
Mode = 6
Find range and IQR for: 3, 5, 7, 9, 11, 13, 15.
Range = 15 − 3 = 12
Q1 = 5, Q3 = 13, IQR = 13 − 5 = 8
Data: 10, 12, 15, 18, 20, 22, 50. Is 50 an outlier?
Q1 = 12, Q3 = 22, IQR = 10
Upper fence = $22 + 1.5(10) = 37$
50 > 37, so yes, 50 is an outlier.
Common Misconceptions
Mean is always the best measure of centre. No — the median is better for skewed data or data with outliers.
Range and IQR are the same. No — range uses all data (max − min), while IQR uses only the middle 50%.
For even $n$, pick either middle value as median. No — average the two middle values.
Practice — Centre and Spread
Economics and Policy
The Australian Bureau of Statistics reports both mean and median household income because the mean is skewed by high earners. The median gives a better picture of the typical Australian family. Similarly, median house prices are reported rather than mean prices.
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▼Mean
- $ar{x} = sum x / n$
- Affected by outliers
Median
- Middle value
- Better for skewed data
Spread
- Range = max − min
- IQR = Q3 − Q1
- IQR ignores outliers