The subject of statistics is broadly divided into two branches: Descriptive Statistics and Inferential Statistics. Descriptive Statistics deals with data collection and summarizing the raw data in an understandable format, and these results are generalized to arrive at a conclusion applying Inferential Statistics. The description or summary of sample behavior, presented using Descriptive Statistics is alone is in turn used for drawing inferences on the Population characteristics. Descriptive Statistics comes into play again in the presentation of the estimated or generalized Population traits.
The data that is summarized in tabular, graphical or numerical form is also known an descriptive statistics.

## Descriptive Statistics Definition

Descriptive Statistics deals with analysis and methods related to collection, organization, summarizing and presentation of data.

Applying the techniques of descriptive statistics, the raw data is collected and transformed into a meaningful form.

The data is summarized using numerical measures of central tendency and dispersion. Different types of graphs and plots provide visual representations of data which aid in understanding the traits and patterns related to the data.

The descriptive statistics summary cannot however be generalized to population traits.

## Purpose of Descriptive Statistics

The purpose of descriptive statistics is not restricted only to meaningful presentation of raw data. Inferential statistics makes use of the numerical summary of the sample data to arrive at a conclusion or generalization. For example, the sample mean calculated is treated using techniques of inferential statistics to estimate population mean, or testing the claim on population mean.

The tabular or graphical presentation of data we find in news papers, magazines or other media are often the generalized results of Sample data. Descriptive statistics is again used in these visual presentations of Population traits.

Thus the two branches of study, descriptive and inferential statistics co exist, complementing each other.

## Descriptive Vs Inferential Statistics

The contrast between Descriptive and Inferential Statistics can be tabulated as follows:

 Descriptive Statistics Inferential Statistics Descriptive Statistics deals with collection, organization,summarizing and presentation of data. Inferential Statistics provides methods for generalizing fromsamples to populations, testing the claims on Population parameters, determining relationship between variables and making predictions. The sample statistics are calculated using formulas andmethods provided in descriptive statistics. Inferential statistics makes use of the sample statistics in estimating the population parameters and testing hypothesis on population parameters. The graphing and charting techniques present in descriptive statistics is used for presentation of Population traits andfor comparing different population behavior. In inferential statistics, graphs are used to study the pattern of trend andto provide a visual prediction for future.

## Descriptive Statistics Examples

Let us solve a problem to show how numerical measures are calculated to summarize data in descriptive statistics. For this purpose we consider data organized in a frequency table as follows:

### Solved Example

Question: In a class of 75 students, a statistics test is conducted for a maximum score of 25. The following table gives the frequencies against the test scores. Find the Mean, Median and Mode for the data.

 Test Score 1 5 6 8 10 12 13 14 15 17 20 21 24 Number of  Students 1 1 2 6 10 16 13 9 8 5 2 1 1

Solution:

The Mean of a frequency distribution is calculated using the formula x = $\frac{\sum fx}{N}$   where N is the total number of observations.

We redo the table making suitable columns to enable the required calculations and include a cumulative frequency column (which is used to determine the median).

 Test Score      x Frequency         f fx CumulativeFrequency 1 1 1 1 5 1 5 2 6 2 12 4 8 6 48 10 10 10 100 20 12 16 192 36 13 13 169 49 14 9 126 58 15 8 120 66 17 5 85 71 20 2 40 73 21 1 21 74 24 1 24 75 ∑f = 75 ∑fx = 943

The numbers in the cumulative frequency column against a test score gives the total count of frequencies for test scores equal and less.

Using the formula for mean

x = $\frac{\sum fx}{N}$  = $\frac{943}{75}$ = 12.6 answer rounded to the tenth.
Hence the average test score = 12.6.

The median is given by the the middle value when the test scores are arranged as an ascending array.  That is $\frac{(75 + 1)}{2}$ = 38th test score is the median. The 38th item will be a test score 13, as 36 items have scores less than 13 and 49 items have scores less than 14.
Thus the median of the data set = 13.

The test score 12 has the greatest frequency = 16. Hence the mode of the data set is 12.