Given:
- Median of 20 observations is 50.
- Mode is also 50.

To find:
- The mean of the observations.

Solution:

The median of a set of observations is the middle value when the observations are arranged in ascending or descending order. Since the median is 50, we can say that the 10th observation is 50.

The mode of a set of observations is the value that appears most frequently. Since the mode is also 50, it means that 50 appears more times than any other value in the set.

Step 1: Setting up the observations

Let's consider the observations arranged in ascending order:
x1, x2, x3, ..., x10, ..., x20

Step 2: Determining the values around the median

Since the median is the 10th observation and we know that 50 is the median, we can write:
x9 < x10="50" />< />

Step 3: Considering the frequency of the mode

Since the mode is also 50, it means that it appears more than once in the set. Let's assume that 50 appears a total of 'm' times.

Step 4: Calculating the sum of the observations

The sum of the observations can be calculated as follows:
Sum = (x1 + x2 + x3 + ... + x9) + (m * 50) + (x11 + x12 + ... + x20)

Step 5: Calculating the mean

The mean is calculated by dividing the sum of the observations by the total number of observations. In this case, there are 20 observations.

Mean = Sum / 20

Step 6: Simplifying the expression

Since we know that the 10th observation is 50, we can replace it in the sum expression:
Sum = (x1 + x2 + x3 + ... + x9) + (m * 50) + (x11 + x12 + ... + x20)
Sum = (x1 + x2 + x3 + ... + x9 + x11 + x12 + ... + x20) + (m * 50)

Since the sum of the observations is the same as the sum of the numbers from x1 to x20, we can write:
Sum = 1 + 2 + 3 + ... + 9 + 11 + 12 + ... + 20 + (m * 50)

Step 7: Simplifying further

The sum of numbers from 1 to n can be calculated using the formula:
Sum = (n * (n + 1)) / 2

Using this formula, we can simplify the expression for the sum as:
Sum = (9 * 10) / 2 + (11 + 12 + ... + 20) + (m * 50)

Step 8: Calculating the mean

Now that we have the sum of the observations, we can substitute it into the mean formula:
Mean = Sum / 20
Mean = [(9 * 10) / 2 + (11 + 12 + ... + 20) + (m * 50)] /