2013
e-GMAT LLC
Shalabh
[ HOW TO COMPARE MEAN
AND MEDIAN IN LESS THAN 20
SECONDS ]
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HOW TO COMPARE MEAN AND MEDIAN IN LESS THAN 20 SECONDS?
Article Title: Compare Mean and Median in less than 20 seconds
In a table analysis question that contains a long list,
calculating mean through traditional method takes
time. To save time, we will introduce two techniques
that will allow you to logically deduce whether mean
is greater than or less than the median.
To illustrate my point, let’s review OG 13/#24
question for discussion.
‘The table lists data on the 22 earthquakes of magnitude 7 or greater on the Richter Scale
during a recent year. Times are given in hours, minutes, and seconds on the 24-hour
Greenwich Mean Time (GMT) clock and correspond to standard time at Greenwich, United
Kingdom (UK). Latitude, measured in degrees, is 0 at the equator, increases from 0 to 90
proceeding northward to the North Pole, and decreases from 0 to –90 proceeding southward
to the South Pole. Longitude, also measured in degrees, is 0 at Greenwich, UK, increases from 0
to 180 from west to east in the Eastern Hemisphere, and decreases from 0 to –180 from east
to west in the Western Hemisphere.’
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For each of the following statements, select Yes if the statement is true based on the
information provided; otherwise select No.
Yes No
24A.
For the 22 earthquakes, the arithmetic mean of the depths is greater than
the median of the depths.
24B.
More than half of the 22 earthquakes occurred north of the equator.
24C.
Exactly half of the earthquakes listed occurred between 10:00:00 and
20:00:00 GMT.
The focus of our discussion will be question # 24A.
STANDARD APPROACH
“For the 22 earthquakes; the arithmetic mean of the depths is greater than the median of the
depths.”
In mathematical terms:
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Is Mean > Median ?
The question asks us to compare mean and median. OUR NATURAL INSTINCT WILL BE
TO FIRST CALCULATE THE MEAN AND THEN THE MEDIAN.
Mean  Mean involves handling 22 data points. Summing up all 22 data points and
dividing the sum by 22 is quite time consuming. The mean comes out to be 112.56.
The median 
1. Median is the value of the middle-most cell of depth column, when data points are
arranged in ascending order.
2. In this dataset, there are 22 elements.
3. So middle most value = mean of 11th and 12th values
4. Median = (25+26)/2=25.5.
Now let’s review the two approaches.
APPROACH 1 – “LIMITED SET OBSERVATION”
Per the approach, we will at first calculate median, since median is less calculation intensive
as compared to mean.
Median = 25.5 (Following the same approach as in “standard approach”)
Mean –
Fundamental principle – In a series of all positive numbers, mean of the series is always
greater than the sum of limited data divided by total # of data points in the set
When we glace at the dataset, we find that the last value 641 is disproportionately high.
This implies that mean of the all depths must be greater than 1/22 times 641. This equals
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to 1/22(641) = 293/22 km. Now, 293/22 itself is already greater than the median depth
(25.5 km), so the actual mean of the depths must be greater than the median of the
depths. So we could arrive at the answer without actually calculating the exact mean of the
list.
This approach has also been used by OG.
APPROACH 2
THE SHERLOCK METHOD - LOGICAL DEDUCTION BY OBSERVATION
This approach is along the lines of the approach 1- Limited set Observation approach, but
there is no calculation involved. In fact this approach relies on data observation only.
Median- Instead of exactly calculating the median, we will make a few observations. After
all to find the exact median, we did have to go through some effort (as shown in standard
approach). We know that median is the middle most value.
Just observing this data set, we can be confident that
the median will fall between 25 and 31.
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Mean- Let us understand a property of mean
in relation to ascending order listed dataset.
The given data set of depth is arranged in
ascending order, & bottom cell values are
disproportionately higher than other cell
values. As all the data points lay equal
importance for mean, this implies that mean
will drift towards heavy values.
So it is obvious that mean value will be more
than the median. In fact it will be much higher
than the median value in this specific question.
Just to recollect the question, “For the 22 earthquakes; the arithmetic mean of the depths is
greater than the median of the depths.” So the answer to the question is YES.
The best thing about this approach is that we don’t need to do any calculation. Mere
observation of data & use of basic concept is enough to determine the answer.
Table given below presents a few salient features of mean and median.
Mean
Importance of set
of observations
Application
1. All data have equal importance.
2. Extreme values can significantly
affect mean.
Yields best representation of average,
when each value is important and
there is less variation in observed
values
Median
1. Middle most values matter the
most.
2. Least affected by extreme
values.
Yields best representation of
average, when few values are too
low and/or too high
OTHER SCENARIOS
1. Few top cell values
are
disproportionately low
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This scenario is opposite of question # 24, we discussed. We can safely infer that mean
would be less than the median, because few top cell values are disproportionally low
compared to other cell values. This will make mean drift towards low values.
The Logical deduction by observation- Sherlock approach will work fine if data points in
other part of the series do not have high variation in values.
2. Few top cell values are disproportionately low and bottom cell values are
disproportionately high
In this scenario, we cannot infer mean is less or more than the median. The top 5 cells’
values will drift mean towards lesser value, whereas bottom 4 cells’ values will drift it
towards higher value. So it is difficult to infer which way mean will drift more.
So what to do in this scenario?
Well, GMAT is not going to test exactly this kind of scenario. Had this scenario been
presented to you in the exam, there would have been some other concept applicable to
deduce the answer. GMAT lays more importance on application of concepts than on long
calculations.

While calculating mean, first arrange the data in ascending order.
o If values in the bottom cells are too high, then Mean > Median
o If values in the top cells are too low then Mean < Median
o If values at the top are too low, & bottom cells are too high, then
do not apply this approach.
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EXERCISE QUESTION
‘13 students from a school were rated for their proficiency in 4 sports - higher the rating, more the
proficiency. Table Tennis and Basketball scores are out of 100 points, Lawn Tennis out of 20 points,
and badminton out of 50 points.’
Students
Table
Tennis
(100)
Lawn
Tennis
(20)
Badminton
(50)
Basket
Ball
(100)
Angio
Gary
Jackson
John
Karren
Kent
Luke
Mathews
Messey
Stacey
Brent
Mary
Joe
85
60
70
65
75
90
80
55
95
7
9
11
21
10
14
6
16
4
18
8
20
12
20
20
19
18
34
36
40
30
48
38
32
44
42
48
49
5
4
36
34
28
32
84
38
30
40
42
65
48
46
36
For each of the following statements, select ‘Yes’ if statement is true based solely on the information
provided in the table; otherwise select ‘No’.
Q1
Q2
Q3
Yes No
o
o Table tennis mean score is more than Basketball mean score.
o
o There are at least 3 sports, whose median scores are greater than their mean scores.
If 2 new students Manish and Imran are included; whose Lawn Tennis scores are 5 & 19
o
o
respectively, then the mean score of Lawn Tennis increases by at least 2 points.