Mean Deviation and Quartile Deviation

Mean deviation and quartile deviation are two measures of dispersion used in statistics. Mean deviation is the average distance of each value from the mean, while quartile deviation is the difference between the third and first quartiles.

Formula for Mean Deviation

The formula for mean deviation is:

Mean Deviation = Σ|xi - x̅| / n

where Σ represents the sum, xi is the i-th value in the data set, x̅ is the mean of the data set, and n is the number of values in the data set.

Formula for Quartile Deviation

The formula for quartile deviation is:

Quartile Deviation = (Q3 - Q1) / 2

where Q1 is the first quartile and Q3 is the third quartile.

Solution

Given that the mean deviation of a normal variable is 16, we can use the formula for mean deviation to find the mean of the data set. Let's assume that the mean is x̅.

Mean Deviation = 16
Σ|xi - x̅| / n = 16

Since we don't know the values of the data set, we can't solve for x̅ directly. However, we can use another property of normal variables: the difference between the third quartile and the mean is approximately 0.67 times the standard deviation.

Q3 - x̅ ≈ 0.67σ

where σ is the standard deviation.

We can use this property to rewrite the formula for mean deviation as:

Mean Deviation = σ * Σ|xi - x̅| / (nσ)
16 = σ * Σ|xi - x̅| / (nσ)

We can simplify this expression by dividing both sides by σ:

16 / σ = Σ|xi - x̅| / (nσ)

Now we can use the property of normal variables to rewrite the left-hand side of the equation:

16 / σ ≈ 0.6745

where 0.6745 is the value of the standard normal distribution corresponding to the 75th percentile.

Substituting this value back into the equation, we get:

0.6745 = Σ|xi - x̅| / (nσ)

Solving for Σ|xi - x̅|, we get:

Σ|xi - x̅| = 0.6745 * nσ

We can substitute this expression back into the formula for mean deviation:

16 = σ * Σ|xi - x̅| / (nσ)
16 = σ * 0.6745 * n

Solving for σ, we get:

σ = 16 / (0.6745 * n)

Now we can use the formula for quartile deviation:

Quartile Deviation = (Q3 - Q1) / 2

We know that Q3 - x̅ ≈ 0.67σ, so:

Q3 - x̅ ≈ 0.67 * 16 / (0.6745 * n)
Q3 - x̅ ≈ 23.728

Similarly, we know that Q1 - x̅ ≈ -0.67σ, so:

Q1 - x̅ ≈ -0.67 * 16 / (0.