GMAT Quant | Statistics Question 10

This GMAT practice question is a very interesting problem solving question from descriptive statistics. It covers the following statistical concepts: Mean, Median, Mode, and Range. A GMAT hard math question.

Question 10: An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month?

1. $4000
2. $13,000
3. $9000
4. $5000
5. $11,000

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Video Explanation

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Explanatory Answer | Mean Median Mode & Range

Hints to solve this GMAT Statistics question

1. Mean and Median = $7000. So, find the third highest incentive.
2. Only one mode; mode = $12,000.
3. Use hint 1 and hint 2 to find how many executives have got $12,000.
4. Now compute the sum of incentives got by those who got neither $7000 nor $12000.

Step 1 of solving this GMAT Statistics Question: Understanding the given data

1. The arithmetic mean of the incentives is $7000.
2. The median of the incentives is also $7000.
3. There is only one mode and the mode is $12,000.

Step 2 of solving this GMAT Statistics Question: Decoding Mean and Median

Let their incentives be a, b, c, d, and e such that a ≤ b ≤ c ≤ d ≤ e

Therefore, the median of these values is 'c'.
The median incentive is $7000. So, c = $7000.

Essentially, the incentives are __ __ 7000 __ __

The arithmetic mean of the incentives is $7000.
So, the sum of their incentives a + b + c + d + e = 5 * 7000 = $35,000

Step 3 of solving this GMAT Statistics Question: Decoding Mode

There is only one mode amongst these 5 observations.
The mode is that value that appears with the maximum frequency.
Hence, $12,000 is the incentive received by the most number of salesmen.

So, the incentives are __ __ 7000, 12000, 12000

Step 4 of solving this GMAT Statistics Question: Putting it all together

The incentive that c has got is $7000
The incentive received by d and e are 12,000 each

Therefore, c + d + e = 7000 + 12,000 + 12,000 = $31,000
Hence, a + b = 35,000 - 31,000 = $4000

As there is only one mode, the incentives received by a and b have to be different.
So, a received $1000 and b received $3000.

Maximum incentive : $12,000
Minimum incentive : $1000
Difference between maximum and minimum incentive : $11,000

Choice E is the correct answer.