Question

A professor marks an exam and, after compilation of the grades,she finds that her 25 students obtained a class mean of 70%. On the other hand, the median is 74% and the mode is 78%.

a) At least how many students scored 74% or more? Explain briefly.

b) What could explain that the mean is less than the median?

c)Is this grade distribution symmetric, right skewed, or left skewed? Explain briefly.

d)One student asks the professor to review his grade. After the review, the student’s grade passes from 77% to 85%.

i)What is the new class mean?Round your answer to 2 decimals.Explain.

ii) What is the new class median? Explain.

Answer 1

a)

Median denotes a value lying at the midpoint of a frequency distribution of observed values or quantities, such that there is an equal probability of falling above or below it.

As the median is 74 so exactly half students scored more than 74 and half students scored less than 74.

As there are 25 students so 1 Student scored 74 (i.e the median) (25-1)/2 = 12 students scored more than 74.

So, 13 students scored 74% or more.

b)

• If the mean is greater than the mode, the distribution is positively skewed.
• If the mean is less than the mode, the distribution is negatively skewed.
• If the mean is greater than the median, the distribution is positively skewed.
• If the mean is less than the median, the distribution is negatively skewed.

c)

The distribution is negatively skewed as Both median and mode are greater than mean.

d)

Previously,

Mean = Sum of marks obtained by all the students / No. of Students = 70

No. of Students = 25

So, Sum of marks obtained by all the students = 70 * 25 = 1750

After the review, the student’s grade passes from 77% to 85%.

$\implies$ Sum of marks obtained by all the students = 1750 + (85-77) = 1758

No. of Students = 25

New Mean = $\frac{1758}{25} = 70.32$

Median Does not change as 74% still remained the middle value i.e at 13th rank as,

Previously 77% was now the right side(Greater side) and after the review also 85% remained at the right side(Greater side).

So, New Median = 74% (No Change)