2.2.2 Box Plots & Cumulative Frequency

Box Plots

Box plots are a visual way of showing the distribution of data. They summarise a data set using five key values:

1. Minimum: The smallest value in the data set.
2. Lower Quartile $(Q1)$: The median of the lower half of the data (25th percentile).
3. Median: The middle value when the data is ordered (50th percentile).
4. Upper Quartile $(Q3)$: The median of the upper half of the data (75th percentile).
5. Maximum: The largest value in the data set.

A box plot is drawn as follows:

• A box is drawn from the lower quartile $(Q1)$ to the upper quartile $(Q3)$.
• A line inside the box represents the median.
• Whiskers extend from the box to the minimum and maximum values. These features allow us to see the spread and skewness of the data at a glance.

Cumulative Frequency

Cumulative frequency is used to find the number of observations below a particular value in a data set. It's especially useful for determining medians, quartiles, and percentiles.

To create a cumulative frequency table:

6. Start with a frequency table showing how often each data point occurs.
7. Add the frequency of each data point to the sum of the frequencies of all previous data points.

Cumulative Frequency Graph

• Plot points at the upper boundary of each data value range against its cumulative frequency.
• Join these points with a smooth curve or straight lines. This graph helps in estimating the median, quartiles, and percentiles directly from the graph.

Connecting Box Plots and Cumulative Frequency:

You can use the cumulative frequency graph to determine key values like the median, Q1, and Q3, which you then use to construct a box plot. This makes these two tools very complementary for analysing data distributions.

Data Analysis of Giraffe Heights

Given Data

The table shows the heights (in meters) of 80 giraffes.

Height, $h$ (m)	Frequency	Cumulative Frequency (C.F.)
4.6 ≤ $h$ < 4.8	$4$	$4$
4.8 ≤ $h$ < 5.0	$7$	$11$
5.0 ≤ $h$ < 5.2	$15$	$26$
5.2 ≤ $h$ < 5.4	$33$	$59$
5.4 ≤ $h$ < 5.6	$17$	$76$
5.6 ≤ $h$ < 5.8	$4$	$80$

Tasks

a) Draw a cumulative frequency diagram.

• A cumulative frequency curve is plotted on the graph.

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b) Estimate the median height of the giraffes.

• Median $≈$ 5.26 m

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c) Estimate the lower quartile and the $90th$ percentile.

• Lower Quartile $(LQ)$ = 5.16 m
• $90th$ Percentile:
  • Calculation: $80 \times 0.9 = 72$
  • $90th$ Percentile $≈ 5.52 m

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d) Draw a box plot to represent this data.

• Min = 4.6 (from the table)

• LQ = 5.16

• Median = 5.26

• UQ = 5.38 (from the graph)

• Max = 5.8 (from the table)

  

• The box plot is drawn using these values.

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e) Estimate the number of giraffes with heights below $5.38$ $m$.

• Since $5.38$ is the upper quartile $(UQ)$, 75% of giraffes have heights below this.
• Calculation: $80 \times 0.75 = :highlight[60]$

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