Measures Of Dispersion

Step 1: Calculate the sum of the five given marks. The given marks are 45, 54, 41, 57, and 43. Their sum is:

45%20%2B%2054%20%2B%2041%20%2B%2057%20%2B%2043%20%3D%20240.

Step 2: Find the sixth test score using the mean. The mean score for the six tests is 48. The sum of all six marks is:

%5Ctext%7BSum%20of%20six%20marks%7D%20%3D%2048%20%5Ctimes%206%20%3D%20288.

Therefore, the sixth test score is:

x_6%20%3D%20288%20-%20240%20%3D%2048.

Step 3: Calculate the variance. The marks are 45, 54, 41, 57, 43, and 48, with the mean

%5Cmu%20%3D%2048

. The variance is calculated as:

%5Ctext%7BVariance%7D%20%3D%20%5Cfrac%7B1%7D%7B6%7D%20%5Csum_%7Bi%3D1%7D%5E%7B6%7D%20(x_i%20-%20%5Cmu)%5E2

Now calculate

(x_i%20-%2048)%5E2

for each mark:
-

(45%20-%2048)%5E2%20%3D%209

(54%20-%2048)%5E2%20%3D%2036

(41%20-%2048)%5E2%20%3D%2049

(57%20-%2048)%5E2%20%3D%2081

(43%20-%2048)%5E2%20%3D%2025

(48%20-%2048)%5E2%20%3D%200

Thus, the variance is:

%5Ctext%7BVariance%7D%20%3D%20%5Cfrac%7B9%20%2B%2036%20%2B%2049%20%2B%2081%20%2B%2025%20%2B%200%7D%7B6%7D%20%3D%20%5Cfrac%7B200%7D%7B6%7D%20%3D%20%5Cfrac%7B100%7D%7B3%7D.

Step 4: Calculate the standard deviation. The standard deviation is the square root of the variance:

%5Ctext%7BStandard%20Deviation%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B100%7D%7B3%7D%7D%20%3D%20%5Cfrac%7B10%7D%7B%5Csqrt%7B3%7D%7D.

Chapter Questions
1 MCQs