Five Number Summary for Even & Odd Number of Data

The five number summary is a set of five quantities associated with a data set that allows us to obtain an overall understanding of the distribution of data values. It consists of the following five pieces of information:

1. Lowest Value.
2. Lower Quartile.
3. Median.
4. Upper Quartile.
5. Highest Value.

The procedure to find the quartiles and the median is different depending on whether the number of data values (denoted by N) is odd or even hence the procedure to obtain five number summary is different in both cases. The lowest and highest values can clearly be found by inspection so we need to explain how to find the median and the quartiles depending on whether N is odd or even. The data must first be arranged in ascending order before applying the procedure explained below.

Five Number Summary for Odd Number of Data Values:

1. If N is odd, the median can be found by the formula, \text{Median }=\left(\frac{N+1}{2}\right )^{th}\text{ term}.
2. The lower and upper quartiles can be obtained using the formulae, Q_1=\left(\frac{N+1}{4}\right )^{th}\text{ term}. Q_3=\left(\frac{3(N+1)}{4}\right )^{th}\text{ term}.
3. If the value (N+1)/4 turns out to be fractional then we can estimate it to the nearest whole integer lying above it.

Example:

Consider the following list of data values: 12, 13, 18, 19, 21, 22, 26, 27, 32, 33, 36, 42, 45.

Here N = 13 (odd). The five number summary is as follows,

1. Lowest Value = 12.
2. Q_1=\left(\frac{N+1}{4}\right )^{th}\text{ term}= \left(\frac{13+1}{4}\right )^{th}\text{ term} = 3.5^{th}\text{ term} = 4^{th}\text{ term} = 19.
3. \text{Median }=\left(\frac{N+1}{2}\right )^{th}\text{ term}= \left(\frac{13+1}{2}\right )^{th}\text{ term} = 7^{th}\text{ term} = 26.
4. Q_3=\left(\frac{3(N+1)}{4}\right )^{th}\text{ term}= \left(\frac{3\times 14}{4}\right )^{th}\text{ term} = 10.5^{th}\text{ term} = 11^{th}\text{ term} = 36.
5. Highest value = 45.

Five Number Summary for Even Number of Data Values:

1. If N is even, the median can be found by the formula, \text{Median }=\frac{1}{2}\left[\left(\frac{N}{2}\right )^{th}\text{ term}+\left(\frac{N}{2}+1 \right )^{th}\text{ term}\right ].
2. The lower and upper quartiles can be obtained using the formulae, Q_1=\left(\frac{N}{4}\right )^{th}\text{ term}. Q_3=\left(\frac{3N}{4}\right )^{th}\text{ term}.

Example:

Consider the following list of data values: 102, 111, 116, 123, 124, 126, 134, 138.

Here N = 8 (even). The five number summary is as follows,

1. Lowest Value = 102.
2. Q_1=\left(\frac{N}{4}\right )^{th}\text{ term}=2^{nd}\text{ term} = 111.
3. \text{Median }=\frac{1}{2}\left[\left(\frac{N}{2}\right )^{th}\text{ term}+\left(\frac{N}{2}+1 \right )^{th}\text{ term}\right ] = \frac{1}{2}[4^{th}\text{ term}+5^{th}\text{ term}] = 123.5.
4. Q_3=\left(\frac{3N}{4}\right )^{th}\text{ term}=6^{th}\text{ term} = 126.
5. Highest value = 138.