# Dispersion of Data: Range, Quartiles, Variance, Standard Deviation, and Interquartile Range

Prerequisite: Basic Statistical Descriptions of Data — Mean, Median, Mode & Midrange

**What do we understand by the Dispersion of Data? **

The dispersion of data means the spread of data.

**How do we measure the dispersion or spread of data?**

Let x1, x2, x3….xn be a set of observations for some numeric attribute, X. We have the following terms for measuring the dispersion of data-

**Range:** It is defined as the difference between the largest and smallest values in the set.

**Quantiles:** These are points taken at regular intervals of data distribution, dividing it into essentially equal-size consecutive sets.

The kth q-quantile for given data distribution is the value x such at most k/q of data values are less than x and at most (q-k)/q of data values are more than x, where k is an integer such that 0 < k < q. Ther are total (q-1) q-quantiles.

For example, the 2-quantile is the data point dividing the lower and upper halves of the data distribution. It corresponds to the median of the set of values. The 4-quantiles are the three data points that split the data distribution into four equal parts, where each part represents one-fourth of the data distribution. *Also commonly called as quartiles*.

**Interquartile Range (IQR):** The distance between the first and third quartiles is a simple measure of the spread that gives the range covered by the middle half of the data. This distance is called the Interquartile range.

IQR = Q3-Q1

**Variance & Standard Deviation:** These are the measures of data dispersion. We can measure how spread out a data distribution is.

A low standard deviation means that the data observations tend to be very close to the mean.

A high standard deviation means that the data are spread out over a large range of values.

The **variance** of N observations, x1, x2, x3….xn, for a numeric attribute X is -

*Mathematically, the standard deviation is defined as the square root of the variance.*