What a teacher!

Probability Level 2

All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

Mode None of these Variance Median All of these Mean

2 solutions

Kay Xspre
Jan 3, 2016

Since Mean, Median and Mode deals with the raw data, for example, from $x_n$ to $x_n+10$ , the three will change accordingly. Variance instead calculates from $x_n-\mu$ , hence if the data increased for 10, this value will be $(x_n+10)-(\mu+10) = x_n-\mu$ , which means the raw value used will remain unchanged, hence the result of Variance unchanged.

Since we are not given the raw data of the original, And therefore the subsequent set, we cannot determine if the variance truly remains unchanged.

"Poor" marks does not give us a basis of comparison for the variance. The variance is undefined for the beginning set, which means that we cannot extrapolate that the gift of grace marks truly makes the variance unchanged.

my suggestion would be to word it "all of the students were given identical poor marks" as it would identify that the students indeed had a zero variance differential.

Joshua Burton - 5 years, 5 months ago

I do not believe the word "poor" is related to the calculation. For the value $x_n$ , if the point are added, the three except Variance are change, and since Variance is based on $\sum_{n=1}^m\frac{(x_n-\mu)^2}{m}$ I fail to see how Variance is undefined, whether the raw or "+10" data

Kay Xspre - 5 years, 5 months ago

Without the answerer knowing that the original marks were identical, we cannot say that the variance does not change once all the grace marks are given.

I agree to where variance comes from, but the ambiguity still provides sufficient doubt that there was zero variance in the original set of scores.

I am not saying that the grace marks have variance, but I am saying that the original wording of the problem leaves a vague area where it is possible that there was even a slight variance.

Joshua Burton - 5 years, 5 months ago

@Joshua Burton – Conversely, the only way Variance will not change is only when all scores are identical, but, if the score are all identical, it will not be possible to determine Mode (as Mode is determined when there are only one or at most two data with the most frequent appearance; if at least 3 are most frequent or the data is not different at all, Mode is then undefined). Hence, I believe the question is not vague with the inclusion of Mode

Kay Xspre - 5 years, 5 months ago

@Kay Xspre – Hello again. After your response I figured there must be some rift in communication between us and I found the culprit. Around the schools and university I went to, "grace marks" were completely new scores given.

Ex.: a student scored 23, the grace mark was 50. The old score completely erased and replaced by the new.

So when I did a little research and found that grace marks are more commonly "add this score to their score," I realized the confusion. I apologize and now completely agree with your solution.

Joshua Burton - 5 years, 5 months ago

What is $\mu$

Department 8 - 5 years, 5 months ago

$\mu$ or $\overline{x}$ denotes Mean.

Kay Xspre - 5 years, 5 months ago

Prince Loomba
Jan 24, 2016

Variance is measure of randomness and increasing 10 marks to each doesnt change randomness

What a teacher!

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