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Interpreting
Data
Mean, Median, and Mode
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Understand the terms mean, median,
mode, standard deviation
Use these terms to interpret data collected
during experiments
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Mean … the average number
Median … the value that lies in the middle
after organizing all the numbers
Mode … the most frequently occurring
number
Which measure of Central
Tendency should be used?
The measure you choose should
give you a good indication of
the typical score in the sample
or population.
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The mean (or average) is found by taking the
sum of the numbers and then dividing by
how many numbers you added together.
The number that occurs most frequently is
the mode.
When the numbers are arranged in numerical
order, the middle one is the median.
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The mean (or average) is found by adding all
the numbers and then dividing by how many
numbers you added together.
Example: 3,4,5,6,7
3+4+5+6+7= 25
25 divided by 5 = 5
The mean is 5
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Notation
x is pronounced ‘x-bar’ and denotes the mean of a
set of sample numbers values
x =
x
n

means add or sum of a set of numbers
x
represents the individual data numbers
n
represents the how many data numbers are in a sample
N
represents the number of data numbers in a population
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
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First add all the grades
together.
The total equals 1061
Now divide 1061 by 13
(total grades in the
class)
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
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First add all the grades
together.
The total equals 1061
Now divide 1061 by 13
(total grades in the
class)
The answer is 81.61
The mean is 81.61
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
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Same thing, just fancy
x is pronounced ‘x-bar’ and denotes the mean of a
set of sample values
x
x =
n
µ
is pronounced ‘mu’ and denotes the mean of all values
µ =
in a population
x
N
Calculators can calculate the mean of data
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The mean is sensitive to every value, so one
exceptional value can affect the mean
dramatically.
The median (next slide) overcomes that
disadvantage.
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The number that occurs most frequently is
the mode.
Example: 2,2,2,4,5,6,7,7,7,7,8
The number that occurs most frequently is 7
The mode is 7
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The mode is the
number which
occurs most
often.
The number
which occurs
most often is
93
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
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The mode is the
number which
occurs most
often.
The number
which occurs
most often is
93
The mode is 93
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
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When numbers are arranged in numerical
order, the middle one is the median.
Example: 3,6,2,5,7
Arrange in order 2,3,5,6,7
The number in the middle is 5
The median is 5
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The median is the
number in the middle
of numbers which are
in order from least to
greatest.
If we count from both
sides the number in
the middle is 83.
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
5/24/2017
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The median is the
number in the middle
of numbers which are
in order from least to
greatest.
If we count from both
sides the number in
the middle is 83.
The median is 83
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Lowest
55
60
75
80
80
80
83
83
93
93
93
93
93
Highest
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If these were your math grades, what
would you learn by analyzing them?
The mean was 81.61. In order to
raise your grades, you would have to
make higher than an 81.61 on the
rest of your assignments.
The mode was 93 which was your
highest grade. You could look at
these papers to see why you made
this grade the most.
The median is a 83. This means that
most of your grades were higher
than your average. Find your weak
area and try to improve.
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
Real Life
◦ Knowing the mean, median, and mode will help you
better understand the scores on your report card. By
analyzing the data (grades) you can find your average,
the grade you received most often, and the grade in
the middle of your subject area.
◦ Better understanding your grades may lead to better
study habits.
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Mean … the most frequently used but is
sensitive to extreme scores
e.g. 1 2 3 4 5 6 7 8 9 10
Mean = 5.5 (median = 5.5)
e.g. 1 2 3 4 5 6 7 8 9 20
Mean = 6.5 (median = 5.5)
e.g. 1 2 3 4 5 6 7 8 9 100
Mean = 14.5 (median = 5.5)
Median
… is not sensitive to extreme scores
… use it when you are unable to use the
mean because of extreme scores
Mode
… does not involve any calculation or
ordering of data
… use it when you have categories (e.g.
occupation)
English
Frequency
300
250
Mean: 54
200
Median: 56
150
Mode: 63
100
50
Mean = 53.78
Std. Dev. = 19.484
N = 4,253
0
0
20
40
60
English
80
100
In everyday life many variables such as
height, weight, shoe size and exam
marks all tend to be normally
distributed, that is, they all tend to look
like the following curve.
Mean, Median, Mode
0.025
0.02
0.015
0.01
0.005
0
0
20
40
60
80
100
It is bell-shaped and symmetrical about the mean
The mean, median and mode are equal
It is a function of the mean and the standard deviation
Measures that indicate the spread of scores:
Range
Standard
Deviation
Range
 It compares the minimum score with the
maximum score
 Max score – Min score = Range
 It is a crude indication of the spread of
the scores because it does not tell us
much about the shape of the distribution
and how much the scores vary from the
mean
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Two distributions may have the same range
but in one distribution the scores may be
tightly packed around the mean while in the
other they may be more spread out from the
mean.
Let’s determine the mean, mode and median
for your individual data
Then we will calculate the population data for
the class!
Standard Deviation
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It tells us what is happening between the
minimum and maximum scores

It tells us how much the scores in the data
set vary around the mean
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It is useful when we need to compare
groups using the same scale
Mean
0.03
= 50
Std Dev = 15
0.025
0.02
0.015
34%
2%
0.01
0.005
34%
2%
14%
14%
0
0
10
20
30
40
50
60
70
80
90
100
scores
5
-3
0%
20
-2
2%
35
-1
16%
50
0
50%
65
+1
84%
80
+2
95
+3
sd
98% 100%
rank
School A
50
10
Mean
S.d.
School B
60
10
School C
70
10
0
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
20
40
60
80
100
120
School A
50
10
Mean
S.d.
School B
50
13
School C
50
16
0
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
20
40
60
80
100
120
National Mean
55
10
Mean
S.d.
School A
60
15
School B
40
15
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-20
0
20
40
60
80
100
120
Best Measure of Center
Advantages - Disadvantages
Table 2-13
Definitions

Symmetric
Data is symmetric if the left half of its
histogram is roughly a mirror of
its
right half.

Skewed
Data is skewed if it is not symmetric
and if it extends more to one side
than the other.
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-13 (a)
SKEWED LEFT
(negatively)
Skewness
Figure
2-13 (b)
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
Figure
2-13 (a)
SKEWED LEFT
(negatively)
Mean
Mode
Median
SKEWED RIGHT
(positively)
Figure
2-13 (c)
The z-score is a conversion of the raw
score into a standard score based on the
mean and the standard deviation.
z-score = Raw score – Mean
Standard Deviation
Example
z-score
Mean = 55
65 – 55
15
= 0.67
Standard Deviation = 15
Raw Score = 65
Use table provided to convert the z-score
into a percentile.
z-score
= 0.67
Percentile = 74.86% (from table provided)
Interpretation: 75% of the group scored
below this score.
Mean
S.d.
National Mean
55
10
School A
60
15
School B
40
15
0.045
Z-score
for Mean of School A = (60 – 55)/10 = 0.2
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
A z-score of 0.2 is equivalent to a percentile of
57.93% on a national basis
Z-score for Mean of School B = (40 – 55)/10 = -1.5
A z-score of –1.5 is equivalent to a percentile of
-20
0
20
40
60
80
100
120
(100-93.32)%,
that is,
6.68%!