Chapter 5 : Measures of Variability
Measures of Variation
We have discussed measure or central tendency to describe the distribution of scores.
However, the measure of central tendency does not uniquely describe a distribution, most
especially if we want to know how close or how far is the distance of the scores of students in a
certain test from the average performance of the group. It is in this line that we make use of
the measures of variation. Measure of variation is a single value that is used to describe the
spread of the scores in a distribution. The term variation that is also known as variability of
dispersion. There are several ways of describing the variation of scores: absolute measures of
variation and relative measures of variation.
Chapter 5 : Measures of Variability
Let us consider the scores
of student in three sections
of mathematics class. We
shall consider the spread of
scores based on graphical
presentation.
Section A
Section B
Section C
12
12
14
15
17
18
18
18
19
23
23
30
xത = 18.25
12
12
12
13
13
14
17
20
20
28
28
30
xത = 18.25
12
12
12
12
12
12
13
26
26
26
26
30
xത = 18.25
SD= 5.15
SD= 6.92
SD = 7.63
Chapter 5 : Measures of Variability
What can you observe about
the mean and the standard
deviation of the three groups of
scores?
Which group of students
performed well in the class?
Which group of scores is most
widespread? Less scattered?
Before answering such questions,
let us first discuss the different
types of measures of variation.
Chapter 5 : Measures of Variability
Types of Absolute Measures of Variation
There are four kinds of absolute variation.
1) Range
2) Inter-quartile and quartile deviation
3) Mean deviation
4) Variance and standard deviation
Chapter 5 : Measures of Variability
Range
Range (R) is the difference between the highest score and the lowest score in a
distribution. Range is the simplest and the crudest measure of variation,
simplest because we shall only consider the highest score and the lowest score.
Range for Ungrouped Data
R = HS – LS
where
R = range value
HS = Highest score
LS = Lowest score
Chapter 5 : Measures of Variability
Example: Find the range of the two groups of score distribution.
Group A
Group B
10(LS)
15(LS)
12
16
15
16
17
17
25
17
Analysis:
26
23
28
25
30
26
35(HS)
30(HS)
The range of Group A = 25 is greater than the
range of Group B = 15. The implication of this is
that scores in group A are more spread out than
the scores in group B or the scores in Group B
are less scattered than the scores in group A.
RA = HS – LS
RB = HS – LS
RA = 35 – 10
RB = 30 – 15
RA = 25
RB = 15
Chapter 5 : Measures of Variability
Range for Grouped Data
R = HSUB – LSLB
where
R = range value
HSUB = upper boundary of the highest score
LSLB = lower boundary of the lowest score
Chapter 5 : Measures of Variability
Example: Find the value of range of the scores of 50 students in Mathematics
achievement test.
x
25 – 32
33 – 40
41 – 48
49 – 56
57 – 64
65 – 72
73 – 80
81 – 88
89 – 97
f
3
7
5
4
12
6
8
3
2
n = 50
LL of LS = 25
LSLB = 24.5
UL of the HS = 97
HSUB = 97.5
R = HSUB – LSLB
R = 97.5 – 24.5
R = 73
Chapter 5 : Measures of Variability
Properties of Range
1. It is quick and easy to understand.
2. It is a rough estimation of variation.
3. It is easily affected by the extreme scores.
Chapter 5 : Measures of Variability
Interpretation of Range Value
When the range value is large, the scores in the distribution are more dispersed,
widespread or heterogeneous. On the other hand, when the range value is small the
scores in the distribution are less dispersed, less scattered, or homogeneous.
Chapter 5 : Measures of Variability
Inter-quartile Range (IQR) and Quartile Deviation (QD)
Inter-quartile range is the difference between the third quartile and the
first quartile.
IQR = Q3 – Q1
Chapter 5 : Measures of Variability
Properties of Inter-quartile Range
1. Reduces the influence of extreme values.
2. Not as easy to calculate as the range.
3. Only consider the middle 50% of the scores in the distribution.
4. The point of dispersion is the median value
Chapter 5 : Measures of Variability
Quartile deviation indicates the distance we need to go above and
below the median to include the middle 50% of the scores. It is based on the
range of the middle 50% of the scores, instead of the entire set.
The formula in computing the value of the quartile deviation is
𝑸𝑫 =
𝑸𝟑 − 𝑸𝟏
,
𝟐
where QD is the quartile deviation value, Q1 is the value of the first quartile
and Q3 us the value of the third quartile.
Chapter 5 : Measures of Variability
Steps in Solving Quartile Deviation
1. Solve for the value of Q1.
2. Solve for the value of Q3.
3. Solve for the value of QD using the formula 𝑸𝑫 =
𝑸𝟑 − 𝑸𝟏
.
𝟐
Chapter 5 : Measures of Variability
Quartile Deviation of Ungrouped Data
𝑸𝟑 − 𝑸𝟏
𝑸𝑫 =
𝟐
Chapter 5 : Measures of Variability
Example: Using the given data 6,8,10,12,12,14,15,16,20, find the quartile deviation.
Solve for Q1.
Solve for Q3.
n=9
𝑄1 =
1
4
𝑛+ 1−
𝑄1 =
1
4
9+ 1−
𝑄1 =
9
4
+
𝑄1 =
12
4
nth score
1
4
nth score
3 nth score
4
nth score
𝑄1 = 3𝑟𝑑 𝑠𝑐𝑜𝑟𝑒
𝑄1 = 10
1
4
𝑄3 =
3
4
𝑛+ 1−
𝑄3 =
3
4
(9) + 1 −
𝑄3 =
27
4
+
𝑄3 =
28
4
nth score
1 nth score
4
𝑄3 = 7𝑡ℎ 𝑠𝑐𝑜𝑟𝑒
𝑄3 = 15
3
4
nth score
IQR = Q3 – Q1
= 15 – 10
3
4
nth score
IQR = 5
𝑸𝟑 − 𝑸𝟏
𝑸𝑫 =
𝟐
15 − 10
=
2
𝑄𝐷 = 2.5
Analysis: The amount that deviates from the
median value is 2.5.
Chapter 5 : Measures of Variability
Example: The data given below
are the scores of fifty (50)
students in Filipino class. Solve
for the value of quartile
deviation (QD).
x
25 – 32
33 – 40
41 – 48
49 – 56
57 – 64
65 – 72
73 – 80
81 – 88
89 – 97
f
3
7
5
4
12
6
8
3
2
n = 50
cf <
3
10
15
19
31
37
45
48
50
Solve for the value of Q1.
Solve for the value of Q3.
Solve for the value of QD
𝑛
4
3𝑛
4
𝑸𝟑 − 𝑸𝟏
𝑸𝑫 =
𝟐
=
50
4
= 12.5
=
3(50)
4
= 37.5
Q1C = 41 – 48
LL = 41
LB = 40.5
cfp = 10
LL = 73
fq = 5
c.i = 8
cfp = 37 fq = 8
𝑄1 = 𝐿𝐵1 +
𝑛
− 𝑐𝑓𝑝1
4
𝑓𝑞1
c.i
𝑄1 = 40.5 +
12.5 − 10
5
𝑄1 = 40.5 +
2.5
5
20
5
𝑄1 = 40.5 + 4
𝑄1 = 40.5 +
𝑄1 = 44.5
8
8
Q3C = 73 – 80
LB = 72.5
𝑄3 = 𝐿𝐵3 +
c.i = 8
3𝑛
− 𝑐𝑓𝑝3
4
𝑓𝑞3
𝑄3 = 72.5 +
37.5 − 37
8
𝑄3 = 72.5 +
0.5
8
𝑄3 = 72.5 +
4
8
𝑄3 = 72.5 + 0.5
𝑄3 = 73.00
8
c.i
8
𝑄𝐷 =
73 − 44.5
2
𝑄𝐷 =
28.5
2
𝑄𝐷 = 14.25
Chapter 5 : Measures of Variability
Interpretation of IQR and QD
The larger the value of the IQR or QD, the more dispersed the scores at
the middle 50% of the distribution. On the other hand, if the IQR or QD
is small, the scores are less dispersed at the middle 50% of the
distribution. The point of dispersion is the median value.
Chapter 5 : Measures of Variability
Analysis for Inter-quartile Range and Quartile Deviation
When the value of IQR and QD is small, the scores are clustered within
middle 50% of the score distribution. On the other hand, the scores are
dispersed in the middle 50% of the distribution when the value of IQR and
QD is large. To determine which group of distribution is more clustered or
disperses you should compare it with another group of distribution since
there is no standard value of a small or large value of IQR and QD.
Chapter 5 : Measures of Variability
Mean Deviation (MD)
Mean deviation measures the average deviation of the values from the
arithmetic mean. It gives equal weight to the deviation of every score in the
distribution.
Mean Deviation for Ungrouped Data
where
MD = mean deviation value
x = individual score
x = sample mean
n = number of cases
𝑀𝐷 =
Ʃȁ𝑥− 𝑥ȁ
𝑛
Chapter 5 : Measures of Variability
Steps in Solving Mean Deviation for Ungrouped Data
1) Solve the mean value.
2) Subtract the mean value from each score.
3) Take the absolute value of the difference in step 2.
4) Solve the mean deviation using the formula
Ʃȁ 𝒙 − 𝒙 ȁ
𝑴𝑫 =
𝒏
Chapter 5 : Measures of Variability
Example: Find the mean deviation of the scores of 10 students in a mathematics test.
Given the scores:35, 30, 26, 24, 20, 18, 18, 16, 15, 10.
Solution:
Ʃ𝑥
x=
𝑛
212
x=
10
x = 21.2
Ʃȁ𝑥 − 𝑥ȁ
𝑀𝐷 =
𝑛
60.4
𝑀𝐷 =
10
𝑀𝐷 = 6.04
Analysis:
The mean deviation of the 10 scores of students
is 6.04. This means that on the average, the value
deviated from the mean of 21.2 is 6.04.
Chapter 5 : Measures of Variability
Mean Deviation for Grouped Data
𝑴𝑫 =
Ʃ𝒇ȁ𝑿𝒎 − 𝒙ȁ
𝒏
where
MD = mean deviation value
f = class frequency
Xm = class mark or midpoint of each category
x = mean value
n = number of cases
Chapter 5 : Measures of Variability
Steps in solving Mean Deviation for Grouped Data
1. Solve for the value of the mean.
2. Subtract the mean value from each midpoint or class mark.
3. Take the absolute value of each difference.
4. Multiply the absolute value and the corresponding class frequency.
5. Find the sum of the results in step 4.
6. Solve for the mean deviation using the formula for grouped data.
Chapter 5 : Measures of Variability
Example 2: Find the mean deviation of the given scores
below.
Solution:
x=
X
10 – 14
15 – 19
20 – 24
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 – 54
f
5
2
3
5
2
9
6
3
5
n = 40
Xm
12
17
22
27
32
37
42
47
52
fXm
60
34
66
135
64
333
252
141
260
Ʃ𝑓 ȁ𝑋𝑚
= 1345
Xm
/Xm − xത/
− xത
-21.63 21.63
-16.63 16.63
-11.63 11.63
-6.63
6.63
-1.63
1.63
3.37
3.37
8.37
8.37
13.37
13.37
18.37
18.37
f/Xm − xത/
108.15
33.26
34.89
33.15
3.26
30.33
50.22
40.11
91.85
Ʃ𝑓/𝑋𝑚 − 𝑥/ҧ =
425.22
Ʃ𝑓𝑋𝑚
𝑛
1345
x=
40
x = 33.63
𝑴𝑫 =
Ʃ𝒇ȁ𝑿𝒎 − 𝒙ȁ
𝒏
𝑴𝑫 =
𝟒𝟐𝟓. 𝟐𝟐
𝟒𝟎
𝑴𝑫 = 𝟏𝟎. 𝟔𝟑
Analysis:
The mean
deviation of the
40 scores of
students is
10.63. This
means that in
the average, the
value deviated
from the mean
of 33.63 is
10.63.
Chapter 5 : Measures of Variability
Variance and Standard Deviation
Variance is one of the most important measures of variation.
It shows variation about the mean.
Population Variance
2
Ʃ(𝑥
−
µ)
ơ2 =
𝑁
Sample Variance
2
Ʃ(𝑥
−
𝑥)
ҧ
𝑠2 =
𝑛−1
Chapter 5 : Measures of Variability
Steps in Solving Variance of Ungrouped Data
1) Solve for the mean value.
2) Subtract the mean value from each score.
3) Square the difference between the mean and each score.
4) Find the sum of the results in step 3.
5) Solve for the population variance or sample variance using the
formula of ungrouped data.
Chapter 5 : Measures of Variability
Example: Using the data below, find the variance and standard deviation of the scores
of 10 students in a science quiz. Interpret the result.
X
19
17
16
16
15
14
14
13
12
10
Ʃx = 146
x = 14.6
mean
14.6
14.6
14.6
14.6
14.6
14.6
14.6
14.6
14.6
14.6
x-𝐱
4.4
2.4
1.4
1.4
0.4
-0.6
-0.6
-1.6
-2.6
-4.6
𝐱)2
(x 19.36
5.76
1.96
1.96
0.16
0.36
0.36
2.56
6.76
21.16
Ʃ(x - 𝐱)2 = 60.40
Population Variance
of Ungrouped Data
2
Ʃ(𝑥
−
µ)
ơ2 =
𝑁
60.40
2
ơ =
10
2
ơ = 6.04
Sample Variance of
Ungrouped Data
2
Ʃ(𝑥
−
𝑥)
ҧ
𝑠2 =
𝑛−1
60.4
2
𝑠 =
9
𝑠 2 = 6.71
Note: If the
standard deviation
is already solved,
square the value
of the standard
deviation to get
the variance.
Chapter 5 : Measures of Variability
Variance of Grouped Data
Population Variance
2
Ʃ𝑓(𝑋
−
µ)
𝑚
ơ2 =
𝑁
Sample Variance
2
Ʃ𝑓(𝑋
−
𝑥)
ҧ
𝑚
𝑠2 =
𝑛−1
Chapter 5 : Measures of Variability
Steps in Solving the Variance of Grouped Data
1) Solve for the mean value.
2) Subtract the mean value from each midpoint or class mark.
3) Square the difference between the mean value and
midpoint or class mark.
4) Multiply the squared difference and the corresponding
class frequency.
5) Find the sum of step 4.
6) Solve the population variance or sample variance using the
formula of grouped data.
Chapter 5 : Measures of Variability
Example: Score distribution of the test results of 40 students in a Filipino class consisting of 50 items. Solve
the variance and standard deviation and interpret the result.
X
f
Xm
fXm
𝐱ത
𝐗 𝐦 − 𝐱ത
(𝐗 𝐦 − 𝐱ത)𝟐
𝐟(𝐗 𝐦 − 𝐱ത)𝟐
15 – 20
21 – 26
27 – 32
33 – 38
39 – 44
45 - 50
3
6
5
15
8
3
n = 40
17.5
23.5
29.5
35.5
41.5
47.5
52.5
141
147.5
532.5
332
142.5
fXm = 1348
33.7
33.7
33.7
33.7
33.7
33.7
-16.2
-10.2
-4.2
1.8
7.8
13.8
262.44
104.04
17.64
3.24
60.84
190.44
787.32
624.24
88.2
48.6
486.72
571.32
Ʃf/Xm − xത)2 =
2 606.4
Population Variance
2
Ʃ𝑓(𝑋
−
µ)
𝑚
ơ2 =
𝑁
2 606.4
2
ơ =
40
ơ2 = 65.16
Sample Variance
2
Ʃ𝑓(𝑋
−
𝑥)
ҧ
𝑚
𝑠2 =
𝑛−1
2 606.4
2
𝑠 =
39
𝑠 2 = 66.83
Chapter 5 : Measures of Variability
Standard deviation is the most important measures of variation. It is also known
as the square root of the variance. It is average distance of all the scores that deviates
from the mean value.
Population Standard Deviation
ơ=
Ʃ(𝑋 − µ)2
𝑁
Sample Standard Deviation
𝑠=
Ʃ(𝑥 − 𝑥)ҧ 2
𝑛−1
Chapter 5 : Measures of Variability
Steps in Solving Standard Deviation of Ungrouped Data
1) Solve for the mean value.
2) Subtract the mean value from each score.
3) Square the difference between the mean and each score.
4) Find the sum of step 3.
5) Solve for the population standard deviation or sample standard deviation
using the formula for ungrouped data.
Note: If the variance is already solved, take the square root of the variance to get the
value of the standard deviation.
Chapter 5 : Measures of Variability
Example: Using the data on the previous example, solve the population and sample
standard deviation.
Population Standard Deviation
ơ=
Ʃ(𝑋 − µ)2
𝑁
ơ=
60.40
10
Sample Standard Deviation
𝑠=
Ʃ(𝑥 − 𝑥)ҧ 2
𝑛−1
𝑠=
60.40
9
ơ = 6.04
𝑠 = 6.71
ơ = 2.46
𝑠 = 2.59
Chapter 5 : Measures of Variability
Steps in Solving the Standard Deviation of Grouped Data
1) Solve for the mean value.
2) Subtract the mean value from each midpoint or class mark.
3) Square the difference between the mean value and midpoint or class mark.
4) Multiply the squared difference and the corresponding class frequency.
5) Find the sum of the results in step 3.
6) Solve for the population standard deviation or sample standard deviation
using the formula for grouped data.
Chapter 5 : Measures of Variability
Population Standard Deviation
ơ=
Ʃ𝑓(𝑋𝑚 − µ)2
𝑁
ơ=
2 606.4
40
ơ = 65.16
Sample Standard Deviation
𝑠=
Ʃ𝑓(𝑋𝑚 − 𝑥)ҧ 2
𝑁−1
𝑠=
2 606.4
39
𝑠 = 66.8308
ơ = 8.07
𝑠 = 8.18
Chapter 5 : Measures of Variability
Interpretation of Standard Deviation
1. If the value of standard deviation is large, on the average, the scores in
the distribution will be far from the mean. Therefore, the scores are
spread out around the mean value. The distribution is also known as
heterogeneous.
2. If the value of standard deviation is small, on the average, the scores in
the distribution will be close to the mean. Hence, the scores are less
dispersed or the scores in the distribution are homogeneous.
Chapter 5 : Measures of Variability
Which group of students
performed well in the class?
Answer: In terms of
performance, the three
sections of students
perform the same because
they have the same mean
value of 18.25.
What can you observe about the mean and the standard deviation of
the three groups of scores?
Answer: The mean of the three groups of scores is the same
which is equal to 18.25 and the standard deviation of section A =
5.15, section B = 6.92, and section C = 7.63.
Chapter 5 : Measures of Variability
Which group of scores is most widespread?
Less scattered? Answer: The standard
deviation of section A = 5.15, section B = 6.92
and section C = 7.63. The scores that are most
scattered are those in section C because they
have the largest value of standard deviation
which is equal to 7.63. On the other hand, the
less scattered group of scores is in section A
which has the smallest value of the standard
deviation which is equal to 5.15. Therefore,
the smaller the value of the standard
deviation on the average the closer the scores
are to the mean value and the larger the value
of the standard deviation on the average
makes the scores scattered from the mean
value. Using the diagram, there are more
scores that are closer to the mean value in
section A than in section B and section C.
Chapter 5 : Measures of Variability
Properties of Variance and Standard Deviation
1. The most commonly used measures of variation most
especially in research.
2. It shows variation of the individual scores about the mean.
Chapter 5 : Measures of Variability
Relative Measure of Variation
Coefficient of variation shows variation relative to the mean. It is used to compare two or
more groups of distribution of scores. Usually expressed in percent, the smaller the value of the
coefficient of variation the more homogeneous the scores in that particular group. On the other
hand, the higher the value of the coefficient of variation the more dispersed the scores in that
particular distribution.
The formula in computing the coefficient of variation is:
where
𝑥ҧ = mean value
s = standard deviation
𝐶𝑉 =
𝑠
𝑥ҧ
100%
Chapter 5 : Measures of Variability
Example: Find the coefficient of
variation of the given data below:
Section A Section B Section C
12
12
14
15
17
18
18
18
19
23
23
30
𝑥ҧ = 18.25
s = 5.15
12
12
12
13
13
14
17
20
20
28
28
30
𝑥ҧ = 18.25
s = 6.92
12
12
12
12
12
12
13
26
26
26
23
30
𝑥ҧ = 18.25
s = 7.63
𝑠
𝐶𝑉𝐴 =
100%
𝑥ҧ
𝑠
𝐶𝑉𝐶 =
100%
𝑥ҧ
5.15
𝐶𝑉𝐴 =
(100%)
18.25
𝐶𝑉𝐶 =
𝑪𝑽𝑨 = 𝟐𝟖. 𝟐𝟐%
𝑪𝑽𝑪 = 𝟒𝟏. 𝟖𝟏%
𝑠
𝐶𝑉𝐵 =
100%
𝑥ҧ
6.92
𝐶𝑉𝐵 =
(100%)
18.25
𝑪𝑽𝑩 = 𝟑𝟕. 𝟗𝟐%
7.63
(100%)
18.25
Analysis: The sores in section A are less
scattered than the scores in section B and
section C. In other words, scores in section A
are more homogeneous than the scores in
section B and Section C. Another way to
interpret this is, the scores in section C are
more spread out than the scores in section A
and section B, or the scores in section C are
more heterogeneous than the scores in
section A and section B.
Chapter 5 : Measures of Variability
Measures of Skewness
Measure of skewness described the degree of departure of the scores from symmetry. The
skewness coefficient SK can be solved using the formula:
SK=
෤
3(𝑥ҧ −𝑥)
𝑠
where 𝑥ҧ = mean value and 𝑥෤ = median value; s = standard deviation.
Skewness can be classified according to the skewness coefficient. If SK > 0, it is called positively
skewed distribution. When SK < 0, it is negatively skewed distribution. However, if Sk = 0, the scores are
normally distributed. The skewness of a score distribution indicates only the performance of the
students but not the reasons about their performance.
Positively skewed or skewed to the right is a distribution where the thin end tail of the graph
goes to the right part of the curve. This happens when most of the scores of the students are below the
mean.
Negatively skewed or skewed to the left is a distribution where the thin end tail of the graph
goes to the left part of the curve. This happens when most scores got by the students are above the
mean.
Chapter 5 : Measures of Variability
In a classroom testing, a positively skewed
distribution means that the students who took
the examination did very poor. Most of the
students got a very low score and only few
students got a high score. Positively skewed
distribution tells you only on the poor
performance of the test takers but not the
reasons why the students did poorly in the
said examination. Poor performance of the
students could be attributed to the following:
ineffective methods of teaching and
instruction, students’ unpreparedness to take
the examination, test items are very difficult,
and there is no enough time to answer the
test item.
Chapter 5 : Measures of Variability
Negatively skewed distribution means
that the students who took the examination
performed well. Most of the scores are high
and there are only few low scores. The shape
of the score distribution indicates the
performance of the students but not the
reasons why most of the students got high
scores. The possible reasons why students got
high score are: the group of students are
smart, there is enough time to finish the
examination, the test are very easy, and there
is an effective instruction and the students
have
prepared
themselves
for
the
examination.
Chapter 5 : Measures of Variability
Example: Find the coefficient of
skewness of the scores of 40 grade 6
pupils in a 100-item test in
Mathematics if the mean is 82 and
the median is 90 with standard
deviation of 15.
Given: 𝑥ҧ = 82
𝑥෤ = 90
s = 15
Analysis:
3(𝑥ҧ − 𝑥)
෤
𝑆𝐾 =
𝑠
𝑆𝐾 =
3(82 − 90)
15
3(−8)
𝑆𝐾 =
15
𝑆𝐾 =
−24
15
𝑆𝐾 = −1.60
The 𝑠𝑘 = −1.60, the value of 𝑠𝑘 is
negative, meaning the score
distribution is negatively skewed.
Most of the scores are high, this
means that the students have
performed excellently in the said
examination.
Chapter 5 : Measures of Variability
Example 2: Find the coefficient of
skewness of the scores of 45
grade 6 pupils in a 150-item test
in Biology, if the mean is 46 and
the median is 40 with standard
deviation of 7.5.
Given:
𝑥ҧ = 46
𝑥෤ = 40
s = 7.5
3(𝑥ҧ − 𝑥)
෤
𝑆𝐾 =
𝑠
3(46 − 40)
𝑆𝐾 =
7.5
3(6)
𝑆𝐾 =
7.5
18
𝑆𝐾 =
7.5
𝑆𝐾 = 2.40
Analysis:
The 𝑠𝑘 = 2.40 , the
value of 𝑠𝑘 is positive, meaning
the score distribution is positively
skewed. Most of the scores are
below the mean. This means that
the students did not perform
well in the said examination.
Chapter 5 : Measures of Variability
Kurtosis
Kurtosis refers to the flatness or peakedness of
one distribution in relation to another.
Types of Kurtosis
Curve A = leptokurtic: K > 3
Curve B = Mesokurtic: K = 3
Curve C = Platykurtic: K < 3