Data, Summary Statistics and Bivariate Data

Starting point: Years 9 and 10

Measures of centre

• Mean $\bar{x}$: add the values and divide by how many there are.
• Median: the middle value when the data is in order. For an even count, average the two middle values.
• Mode: the most common value.

The median is not affected much by an extreme value, so it is better for skewed data such as house prices.

Measures of spread

• Range = highest $-$ lowest.
• Quartiles: $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.
• Interquartile range $IQR=Q_3-Q_1$, the spread of the middle 50%.

Box plots

Outliers are values below $Q_1-1.5\times IQR$ or above $Q_3+1.5\times IQR$. Here the fences are $-6$ and 26, so there are no outliers.

Standard deviation

The standard deviation $\sigma$ measures the typical distance of the data from the mean. Cart A has a small standard deviation. Cart B has a large one. Calculators find it in statistics mode.

Shape

• Symmetric: mean $\approx$ median.
• Positively skewed: a long tail to the right, mean pulled above the median.
• Negatively skewed: a long tail to the left.

Quick summary

• Mean, median and mode measure the centre. Range, IQR and standard deviation measure spread.
• Five-number summary: minimum, $Q_1$, median, $Q_3$, maximum. Draw it as a box plot.
• Outliers lie beyond $Q_1-1.5\times IQR$ or $Q_3+1.5\times IQR$.
• Skewed data pulls the mean towards the tail. Use the median there.
• For two variables: scatter plot, correlation $r$ and a least-squares line.

Quiz