60, 72, 64, 67, 70, 68, 71, 68, 73, 59.

The mean height for this group is

(1/10)*(60+72+64+67+70+68+71+68+73+59) = 670/10 = 67.2.

First order the data:

59, 60, 64, 67, 68, 68, 70, 71, 72, 73.

Since there are 10 observations, the median is the average of the 5th and 6th observations,
which in this case are identical:

5th observation = 68, 6th observation = 68, median = 68.

The distance between the first and third quartiles is known as the **Inter-Quartile Range**
(IQR).

A useful graphical representation of a distribution including the quartiles is a boxplot.

First order the data:

59, 60, 64, 67, 68, 68, 70, 71, 72, 73.

Since there are an even number of observations (10), the first half of the data is considered in calculating the first quartile:

59, 60, 64, 67, 68.

The median of these values is 64, so this is the first quartile.

The second half of the data is considered in calculating the third quartile:

68, 70, 71, 72, 73.

The median of these values is 71, so this is the third quartile.

For this example, the Inter-Quartile Range is 71-64 = 7.

This MINITAB boxplot corresponds to the student height data. The quartiles have been
calculated by MINITAB to represent levels for 25% and 75% of the data, with
resulting values of 63 and 71.25, respectively (see the note below for details on this calculation).
No outliers have been identified in this boxplot, since all of the observations are within
1.5*IQR from the upper and lower
quartiles.

** Note:** To calculate the quartiles more precisely, first multiply the percentage of interest

2nd observation = 60, 3rd observation = 64.

1st quartile = 0.25*60 + 0.75*64 = 15 + 48 = 63.

The third quartile may be calculated similarly: 0.75*11 = 8.25, so the upper quartile lies between the 8th and 9th observation. The remainer is equal to 0.25, so 25% of the weight is placed on the 9th observation and 75% of the weight is placed on the 8th observation.

8th observation = 71, 9th observation = 72.

3rd quartile = 0.75*71 + 0.25*72 = 53.25 + 18 = 71.25.

- s² = 1/9[(59-67.2)² + (60-67.2)² +
64-67.2)² + (67-67.2)² + .... + (73-67.2)²]

- = 1/9[67.24 + 51.84 + 9.4 + 0.04 + .... + 33.64]

- = 1/9[208.76]

- = 23.2

- The standard deviation is the square root:
*s*= 4.8.

The MINITAB "DESCRIBE" command provides a numerical summary for data which includes the mean, median, standard deviation (abbreviated StDev), minimum and maximum values (Min and Max), and the first and third quartiles (abbreviated Q1 and Q3). The output for our student height example is shown below:

Descriptive Statistics Variable N Mean Median Tr Mean StDev SE Mean C1 10 67.20 68.00 67.50 4.83 1.53 Variable Min Max Q1 Q3 C1 59.00 73.00 63.00 71.25