The following observations have been written in the ascending order :

28, 31, 47, 51, a, a + 3, 71, 74, 78, 90

If the median of the data is 61.5, then the mean of the data is :

The correct answer is 59.3.

Step 1: Understand the given data
The given observations in ascending order are:
28, 31, 47, 51, a, a + 3, 71, 74, 78, 90

The total number of observations, N = 10 (which is an even number).

The median of the data is given as 61.5.

Step 2: Find the value of a using the formula for median
For an even number of observations N, the median is the arithmetic mean of the (^N/₂)^th term and the (^N/₂ + 1)^th term.

Here, ^N/₂ = 5, so the median is the average of the 5^th and 6^th observations.

5^th observation = a
6^th observation = a + 3

Substituting the values into the median formula:

$\text{Median}=\frac{\text{5th observation}+\text{6th observation}}{2}$

$61.5=\frac{a+(a+3)}{2}$

$61.5\times 2=2a+3$

$123=2a+3$

$2a=120$

$a=60$

Step 3: List all observations
Now we can substitute a = 60 into the dataset:
5^th observation = 60
6^th observation = 60 + 3 = 63

The complete set of observations is:
28, 31, 47, 51, 60, 63, 71, 74, 78, 90

Step 4: Calculate the mean of the dataset
The mean is calculated using the formula:

$\text{Mean}=\frac{\text{Sum of all observations}}{\text{Total number of observations}}$

First, find the sum of all observations:

$\text{Sum}=28+31+47+51+60+63+71+74+78+90=593$

Now, calculate the mean:

$\text{Mean}=\frac{593}{10}=59.3$

Thus, the mean of the data is 59.3.