Mean, Median, and Mode:

Mean, median, and mode are the three measures of central tendency. They are used to describe the center of a set of data.

- Mean: The mean is the average of all the numbers in a dataset. It is calculated by summing the values and dividing by the total number of values.
- Median: The median is the middle value in a dataset. It is the value that separates the lower half from the upper half of the data.
- Mode: The mode is the value that appears most frequently in a dataset.

Difference between Mean and Mode:

The difference between mean and mode is a measure of the degree of skewness in a distribution. If the mean is greater than the mode, the distribution is said to be positively skewed. If the mode is greater than the mean, the distribution is said to be negatively skewed.

If the difference between mean and mode is 63, we can conclude that the distribution is positively skewed.

Difference between Mean and Median:

The difference between mean and median is a measure of the degree of symmetry in a distribution. If the mean and median are equal, the distribution is said to be symmetric. If the mean is greater than the median, the distribution is said to be positively skewed. If the median is greater than the mean, the distribution is said to be negatively skewed.

To find the difference between mean and median, we need to know the values of mean and median. We can use the following formula to calculate the median:

Median = (n + 1) / 2

where n is the total number of values in the dataset.

Solution:

Let us assume that the mean of the dataset is M and the mode is Mo. We are given that the difference between mean and mode is 63. Therefore, we can write:

M - Mo = 63

We need to find the difference between mean and median. Let us assume that the median is Md. We know that the number of values in the dataset is odd because we need to find the middle value to calculate the median.

Let us assume that there are 2k + 1 values in the dataset, where k is an integer. Therefore, we can write:

Md = (2k + 2) / 2 = k + 1

We know that the sum of all the values in the dataset is:

Sum = M x (2k + 1)

We can also write:

Sum = Mo x (2f - 1)

where f is the frequency of the mode.

We can substitute the value of Mo from the first equation into the second equation:

Sum = (M - 63) x (2f - 1)

We can simplify this equation:

M x (2k + 1) = (M - 63) x (2f - 1)

2Mk + M = 2Mf - 63f + f

2Mk + M = 2Mf - 62f

2M(k - f) = 62f - M

M = (62f - 2M(k - f)) / 2

We can substitute the value of M from this equation into the first equation:

(62f - 2M(k - f)) / 2 - Mo = 63

31f - M(k - f) - Mo = 63

31f - (62f - 2M(k - f