# Observations of a raw data are 5, 28, 15, 10, 15, 8, 24. Add four more numbers so that mean and median of the data remain the same, but mode increases by 1.

Given, data is 5, 28, 15, 10, 15, 8, 24

No of terms,

We know,

Arithmetic mean of x1, x2, x3, …, xn (n observations) is

Therefore, mean of above data

For finding median, let's arrange data in increasing order

5, 8, 10, 15, 15, 24, 28

As, n is even

Median is term.

Median is 4th term

median = 15

Also, mode is the most frequently occurring value.

Clearly, 15 occurs two times and is most frequent

Therefore, mode is 15.

Now, we have to add four terms to this data such that mean and median remains same, but mode increase by 1.

So, mode of new data = 16

But as 16 is mode, its frequency must be greater than frequency of 2 i.e. minimum 3.

So, we have three terms out of four as 16, 16 and 16.

Now, let 4th term be 'x'.

New data,

5, 8, 10, 15, 15, 16, 16, 16, 24, 28, x

No of terms = 11

As, mean remains same, mean of above data = 15

By using formula, mean of above data

165 = 153 + x

x = 12

So, four terms are 12, 16, 16, 16.

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