Data lists

This looks at data in the raw, ungrouped form.

Contents

• Quartiles and percentiles
• Outliers
• Box and whisker diagram
• Note
• Calculator
• Mean ($\bar x$)
• HL: Variance ($s^2$) and standard deviation ($s$) by hand

Quartiles and percentiles

They increase as data increase. Data is greater has higher quartiles and percentiles.

Quartiles should be found using graphing calculator, which can also sort the data.

Outliers

Outliers lies more than $1.5 \times$ the inter-quartile range (IQR) away from the nearest quartile.

Outliers don’t necessarily need to be thrown away. They could mean measuring a value that has a very small chance of occurring, in addition to indicating a potential mistake. As such, presence of an outlier does not require recomputing any statistical value or measure, unless otherwise specified in the question.

Box and whisker diagram

It draws a box from $Q1$ to $Q2$ and from $Q2$ to $Q3$, and lines (whiskers) from minimum to $Q1$ and from $Q3$ to maximum.

When there are outliers, change the whiskers to the min and max values that are not outliers, without changing the boxes.

Note

Mode and mean cannot be determined from a box and whisker plot; instead they require access to the original, individual values.

Calculator

Be able to input a list of unordered numbers and report the

• mean
• median
• mode
• quartiles (including median)
• variance
• standard deviation

Mean ($\bar x$)

HL: Variance ($s^2$) and standard deviation ($s$) by hand

The formula for variance is

Note that the $-{\bar x}^2$ is outside the summation. Taking the positive square root of both sides is the formula for standard deviation.

A derivation of this formula is provided here.