**Find mean median and mode of grouped data :**

Here we are going to see how to find mean median and mode of grouped data.

**Mean :**

Arithmetic mean (AM) is one of the measures of central tendency which can be defined as the sum of all observations divided by the number of observations.

**Median :**

Median is defined as the middle value of the data when the data is arranged in ascending or descending order.

**Mode :**

**If a set of individual observations are given, then the mode is the value which occurs most often.**

Let us look into some example problems to understand how to find mean, median and mode of the grouped data.

**Example 1 :**

Find the mean, median and mode for the following frequency table:

x 10 20 25 30 37 55 |
f 5 12 14 15 10 4 |

**Solution :**

Arithmetic mean = ∑fx / N

x 10 20 25 30 37 55 |
f 5 12 14 15 10 4 N = 60 |
fx 50 240 350 450 370 220 ∑fx = 1680 |

Arithmetic mean = ∑fx / N = 1680 / 60

= 28

Hence the required arithmetic mean for the given data is 28.

**Median :**

x 10 20 25 30 37 55 |
f 5 12 14 15 10 4 |
Cumulative frequency 5 5 + 12 = 17 17 + 14 = 31 31 + 15 = 46 46 + 10 = 56 56 + 4 = 60 |

Here, the total frequency, N = ∑f = 60

N/2 = 60 / 2 = 30

The median is (N/2)^{th} value = 30^{th} value.

Now, 30^{th} value occurs in the cumulative frequency 31, whose corresponding x value is 25.

Hence, the median = 25.

**Mode :**

By observing the given data set, the number 30 occurs more number of times. That is 15 times.

Hence the mode is 30.

Mean = 28

Mode = 25 and

Mode = 30.

**Example 2 :**

Find the mean, median and mode for the following frequency table:

x 19 21 23 25 27 29 31 |
f 13 15 20 18 16 17 13 |

**Solution :**

To find arithmetic mean for this problem, let us use assumed mean method.

Here A = 25

x 19 21 23 25 27 29 31 |
f 13 15 20 18 16 17 13 N = 112 |
d = x - A -6 -4 -2 0 2 4 6 |
fd -78 -60 -40 0 32 68 78 ∑fd = 0 |

Arithmetic mean = A + [∑fd / N]

= 25 + (0/112)

= 25 + 0

= 25

Hence the required arithmetic mean for the given data is 25.

**Median :**

x 19 21 23 25 27 29 31 |
f 13 15 20 18 16 17 13 |
Cumulative frequency 13 13 + 15 = 28 28 + 20 = 48 48 + 18 = 66 66 + 16 = 82 82 + 17 = 99 99 + 13 = 112 |

Here, the total frequency, N = ∑f = 112

N/2 = 112 / 2 = 61

The median is (N/2)^{th} value = 61^{th} value.

Now, 61^{th} value occurs in the cumulative frequency 25, whose corresponding x value is 25.

Hence, the median = 25.

**Mode :**

By observing the given data set, the number 23 occurs more number of times. That is 20 times.

Hence the mode is 23.

Mean = 25

Mode = 25

Mode = 23.

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