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**INTRODUCTION**

In various fields, we need information in the form of numerical figures called **data.**

These data may relate to the marks obtained by the pupils of a class in a certain examination; the weights, heights, ages, etc., of pupils in a class; the monthly wages earned by workers in a factory; the population of a town or the profits of a company during last few years, etc.

Evaluation of such data helps analysts study the various growth patterns and formulate future targets or policies or derive certain inferences.

**STATISTICS**

It is the science which deals with the collection, presentation, analysis and interpretation of numerical data.

In singular form, statistics is taken as a subject. And, in plural form, statistics means data.

**DATA**

The word data means a set of given facts in numerical figures.

**Fundamental Characteristics of Data**

(i) Numerical facts alone form data. Qualitative characteristics, like honesty, poverty, etc., which cannot be measured numerically do not form data.

(i) Data are aggregate of facts. A single observation does not form data.

(ii) Data collected for a definite purpose may not be suitable for another purpose.

**Types of Data**

**(i) Primary Data :** The data collected by the investigator himself with a definite plan in mind are known as primary data.

**(ii) Secondary Data :** The data collected by someone, other than the investigator, are known as secondary data.

**VARIABLE**

A quantity which can take different values is called a **variable.**

**Ex :** Height, Age and Weight of pupils in a class are three variables.

If we denote them by x, y and z respectively, then values of x give the heights of the pupils; the values of y give the ages of the pupils and the values of z give the weights of the pupils.

**Continuous and Discrete Variables**

Variables are of 2 types

**1. Continuous Variable :** A variable which can take any numerical value within a certain range is called a continuous variable.

**Ex.**

(i) Wages of workers in a factory

(ii) Heights of children in a class

(iii) Weights of persons in a group etc.

**2. Discontinuous (or Discrete) Variable **: A variable which cannot take all possible values between two given values, is called a discontinuous or discrete variable.

**Ex. **

(i) Number of members in a family

(ii) Number of workers in a factory

Such variables cannot take any value between 1 and 2, 2 and 3, etc.

**IMPORTANT TERMS**

**Range :** The difference between the maximum and minimum values of a variable is called its range.

**Variate :** A particular value of a variable is called variate.

**Presentation of Data :** Putting the data in condensed form in the form of a table, is known as presentation of data.

**Frequency :** The number of times an observation occurs is called its frequency.

**Frequency Distribution :** The tabular arrangement of data showing the frequency of each observation is called its frequency distribution.

**ARITHMETIC MEAN**

The average of numbers in arithmetic is known as the Arithmetic Mean or simply the mean of these numbers in statistics.

**MEAN OF UNGROUPED DATA**

The mean of n observations x1, x2, ...., xn is given by

where the symbol , called sigma stands for the summation of the terms.

**Ex.14 The heights of 6 boys in a group are 142 cm, 154 cm, 146 cm, 145 cm, 151 cm and 150 cm. Find the mean height per boy. Sol**.

**Ex.15 Find the mean of the first five multiples of 7.**

Sol. The first five multiples of 7 are 7, 14, 21, 28 and 35.

**MEAN FOR AN UNGROUPED FREQUENCY DISTRIBUTION**

**I. Direct Method**

Let n observations consist of values x_{1}, x_{2}, ...., x_{n }of a variable x, occurring with frequencies f_{1}, f_{2}, .... , f_{n} respectively.

Then, the mean of these observations is given by :

**Ex.18 The ages of 40 students in a class are given below :**

**Find the mean age of the class.**

**Sol.** We prepare the table as given below :

Age (in years) Number of students 1;, xi

**MEDIAN OF UNGROUPED DATA**

**Median : **After arranging the given data in an ascending or a descending order of magnitude, the value of the middle-most observation is called the median of the data.

**Method for Finding the Median of An Ungrouped Data**

Arrange the given data in an increasing or decreasing order of magnitude. Let the total number of observations be n.

(i)

(ii)

**Ex.21 The marks of 13 students (out of 50) in an examination are:**

**39, 21, 23, 17, 32, 41, 18, 26, 30, 24, 27, 36, 9. Find the median marks.**

**Sol.** Arranging the marks in an ascending order, we have :

9, 17, 18, 21, 23, 24, 26, 27, 30, 32, 36, 39, 41

Here, n = 13, which is odd .

= Value of 7th term = 26.

Hence, the median marks are 26.