To make all the study of data useful, we need to use measures of central tendencies. Some of the tendencies are

1. Mean

The mean is the average of the number of observations. It is calculated by dividing the sum of the values of the observations by the total number of observations.

It is represented by bar or .

The mean of n values is given by

**Mean of Grouped Data (Without Class Interval)**

If the data is organized in such a way that the frequency is given but there is no class interval then we can calculate the mean by

where, are the observations

are the respective frequencies of the given observations.

**Example**

Here, , and x_{5} are respectively.

and are respectively.

2. Median

The median is the middle value of the given number of the observation which divides into exactly two parts.

For median of ungrouped data, we arrange it in ascending order and then calculated as follows

- If the number of the observations is odd then the median will be

As in the above figure the no. of observations is 7 i.e. odd, so the median will be term.

- If the number of observations is even then the median is the average of and term.

3. Mode

The mode is the value of the observation which shows the number that occurs frequently in data i.e. the number of observations which has the maximum frequency is known as the Mode.

**Example**

Find the Mode of the following data:

15, 20, 22, 25, 30, 20,15, 20,12, 20

**Solution**

Here the number 20 appears the maximum number of times so

Mode = 20.

The empirical relation between the three measures of central tendency is

3 Median = Mode + 2 Mean

**Question 1: The number of family members in 10 flats of society are**

**2, 4, 3, 3, 1, 0, 2, 4, 1, 5.**

**Find the mean number of family members per flat.**

**Solution**:

Number of family members in flats –

So, we get,

**Question 2 .The following is the list of number of coupons issued in a school canteen during a week:**

**105, 216, 322, 167, 273, 405 and 346.**

**Find the average no. of coupons issued per day.**

**Solution:**

Number of coupons issued in a week: and .

So, we get,

**Question 3.If the mean of six observations y, y + 1, y + 4, y + 6, y + 8, y + 5 is 13, find the value of y.**

**Solution:**

**Question 4.** **The mean weight of a class of 34 students is 46.5 kg. If the weight of the new boy is included, the mean is rises by 500 g. Find the weight of the new boy.**

**Solution:**

The mean weight of students

Sum of the weight of students

Change or increase in the mean weight when the weight of a new boy is added

So, the new mean

So, let the weight of the new boy be y.

So,

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