Data Analysis
Calculate the five-number summary, construct and interpret box plots, and identify outliers in data sets.
The Five-Number Summary
The five-number summary describes a data set using five key values that capture the spread and centre of the data.
Min
Smallest value
Q1
Lower quartile (25th percentile)
Median
Middle value (Q2)
Q3
Upper quartile (75th percentile)
Max
Largest value
Interquartile Range (IQR): IQR = Q3 − Q1. This measures the spread of the middle 50% of the data and is resistant to outliers.
Box Plots (Box-and-Whisker Plots)
A box plot is a graphical display of the five-number summary. It shows the distribution and spread of data at a glance, making it easy to compare two or more data sets.
Box plot with Min=5, Q1=15, Median=22, Q3=35, Max=45. IQR = 35 − 15 = 20.
Identifying Outliers
An outlier is a data value that is significantly different from the rest of the data. The standard rule uses the IQR to define boundary fences:
Lower Fence
Q1 − 1.5 × IQR
Upper Fence
Q3 + 1.5 × IQR
Any data value below the lower fence or above the upper fence is classified as an outlier. Outliers are plotted separately on a box plot as individual points.
Key Vocabulary
| Term | Definition |
|---|---|
| Quartile | Q1, Q2 (median), Q3 divide an ordered data set into four equal parts. |
| Interquartile Range (IQR) | IQR = Q3 − Q1. Measures the spread of the middle 50% of the data. |
| Box plot | A diagram showing the five-number summary with a box (Q1 to Q3) and whiskers to Min and Max. |
| Outlier | A data value more than 1.5 × IQR below Q1 or above Q3. |
Worked Examples
Finding the Five-Number Summary
Data: 4, 7, 10, 12, 15, 18, 22, 25, 30
Min = 4, Max = 30
Median (Q2) = 15 (5th value from 9)
Q1 = median of lower half {4, 7, 10, 12} = (7+10)/2 = 8.5
Q3 = median of upper half {18, 22, 25, 30} = (22+25)/2 = 23.5
IQR = 23.5 − 8.5 = 15
Checking for Outliers
For the data above (Q1 = 8.5, Q3 = 23.5, IQR = 15), is 55 an outlier?
Upper fence = Q3 + 1.5 × IQR = 23.5 + 22.5 = 46
Lower fence = Q1 − 1.5 × IQR = 8.5 − 22.5 = −14
55 > 46 (upper fence), so 55 IS an outlier.
Reading a Box Plot
A box plot shows: Min=10, Q1=25, Median=40, Q3=55, Max=70. What is the IQR and what percentage of data lies between Q1 and Q3?
IQR = Q3 − Q1 = 55 − 25 = 30
50% of the data lies between Q1 and Q3 (the box represents the middle 50%).
The whiskers each cover approximately 25% of the data.
Knowledge Check
Loading questions…
Key Concepts Summary
- ●The five-number summary is: Min, Q1, Median, Q3, Max.
- ●IQR = Q3 − Q1. It measures the spread of the middle 50% of the data.
- ●A box plot displays the five-number summary graphically with a box (Q1–Q3) and whiskers.
- ●Outliers lie beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR and are plotted as separate points.
- ●The box contains 50% of the data; each whisker covers approximately 25%.