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Interquartile Range and A Boxplot To Describe A Distribution

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Interquartile range:

Interquartile range, a measure of descriptive statistics containing middle 50 percent of a data set is an estimation of statistical dispersion. The measure of interquartile range equals to the difference between third and first quartile also known as 75th and 25th percentile of the distribution respectively. Interquartile range provides a measure reading location of most of the observation of a distribution. The estimation of interquartile range provides a measure of variability of a particular data set in terms of dividing the data set into quartiles. Quartiles again divide the data set according to the rank into four equal parts. The values assigned to separate parts are called first, second and third quartiles. The three quartiles are respectively denoted as Q1, Q2 and Q3. As the interquartile range covers bulk of observation in a data set, it is generally preferred to many other measures of dispersion or spread like mean or median. The interquartile range is first quartile subtracted from the third quartile.

 

Box plot of interquartile range

The quartile values can be presented using a box plot. A box plot refers to a graph that provides a good indication regarding spread of observation in the dataset. The interquartile range abbreviated as IQR indicates width of the box in the box plot. In order to find the interquartile range and present in a box plot following steps needs to be followed

Step 1: Sort the data

The first step in drawing a box plot is to arrange the data set. The data set needs to be arranged from smallest to the largest. This is similar to that is done before computing median where data are arranged in ascending order.

Step 2: Compute the median

After sorting the data in ascending order, the next step is to compute the median value. Median value comprises center value of the data set. If the data contains odd number of observation, median is the centermost number. In case, the data has even number of observations, median is the arithmetic average of two centermost observations. The average is estimated by first summing the two numbers and then divide the sum by 2.

Step 3: Upper and lower medians

Once the median for the complete data set is obtained, the next step is to derive the median values of upper and lower portion for the data. When the data set contains odd number of observation, the centermost value or median of the data set needs to be omitted. After that median values of upper and lower portion of the data set are computed.

Step 4: Calculate the difference

Lastly, the difference between upper and lower medians needs to be computed by subtracting lower median from the upper median. This gives the desired value of interquartile range.

 

Figure 1: Box plot and interquartile range

Five number summary and box plot of a distribution

The five number summary measure belong to the set of descriptive statistics that provide information such as variability, nature of distribution and such other related to the data set. The five number summary measure include five most important percentile of the data set. The five measures are the sample minimum, the first quartile or lower quartile, median, the third quartile or upper quartile and the sample maximum. Each of these five measures are described below.

Sample minimum: The sample minimum measure in statistics refers to the smallest observation in the sample data set. This is the least element of the sample. This is one of basic summary statistics in the descriptive statistics measure.

First quartile: The lower quartile is the middle number located between median and the smallest number of the data set. The first quartile or lower quartile indicates 25th percentile of the data set. It splits off the lowest 25 percent observation of the sample data from the highest 75 percent.

Median: Median refers to the second quartile of the data set.  It is the value that separates higher half of the sample data from that of the lower half. For a given data set, median is known as the middle value of the data.

Third quartile: The third quartile is the middle number located between median and the highest number of the data set. The upper quartile or third quartile indicates 75th percentile of the data set. It splits off the highest 25 percent observation of the sample data from the lowest 75 percent.

Sample maximum: The sample maximum is the largest observation of the sample data. The sample maximum indicates greatest element of the data.

Box plot consisting the five summary measure is standard method of presenting distribution of the dataset. In the simplest form of box plot, there is a central rectangle expanding from the lower quartile (first quartile) to the upper quartile (third quartile). The rectangle actually indicates the interquartile range. Inside the segment, there lies median of the data set. The sample maximum and the sample minimum are located above and below the box respectively. A box plot consisting five summary measures looks as follows

 

Figure 2: Five summary measure and box plot

The box plot attempts to display full range of variation ranging from minimum to maximum of the data base. From the box plot, outliers are identified as observations that are either 3×IQR or more lying above the third quartile or 3×IQR or more lying below the first quartile.

A box plot helps to identify whether a given dataset is symmetric or not. In case, the data follows a symmetric distribution, then it is more or less same on both side if cut down from the middle of the box. In case the distribution is skewed a lopsided box is obtained. In this case, median divides the box into unequal parts. If longer part lies to the right, then it is positively skewed. If longer part lies to the left, then it is negatively skewed.

 

Figure 3: Symmetric distribution

 

Figure 4: Left skewed distribution

 

Figure 5: Right skewed distribution

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