The document discusses kurtosis, which refers to the peakedness of a distribution. It presents formulas for calculating kurtosis from both ungrouped and grouped data. As an example, it computes the kurtosis from a set of ungrouped data and determines that the distribution is platykurtic, with a kurtosis value less than 3. It also computes the kurtosis from a set of grouped data and again finds the distribution to be platykurtic.

1. Measure of Kurtosis
Presented by:
Group 2

2. O b j e c t i v e s :
1. Discuss what Kurtosis is.
2. Compute the Kurtosis for both the
grouped and ungrouped data.
3. Be able to describe the Kurtosis
measurement.

3. 1st type Leptokurti
c
Kurtosis - refers to the degree to which a
distribution is peaked or flat.
3 types of
Kurtosis
2nd
type
3rd type
Mesokurtic
Platykurtic
It has less
kurtosis than the
normal
distribution.
The kurtosis of
this distribution is
comparable to
that of the normal
distribution.
It has more
kurtosis than the
normal
distribution.
K < 3 K = 3 K > 3

4. 3 types of Kurtosis

5. Formulas
used in
Ungrouped
data of
Kurtosis

6. 𝑿 =
𝑿
𝒏
Where:
Xi = Individual Values
𝑋= Mean
n = Number of
Observations
s = Standard Deviation
Mean:
For Ungrouped Data:
𝒔 =
(𝑿𝒊 − 𝑿)𝟐
𝒏 − 𝟏
𝑲 =
(𝑿𝒊 − 𝑿)𝟒
𝒏𝒔𝟒
Standard
Deviation:

7. Solve this!

8. Ungrouped Kurtosis
Xi
6
7
9
11
13
15
16
Σ = 77
𝑋 =
𝑋
𝑛
𝑋 =
6 + 7 + 9 + 11 + 13 + 15 + 16
7
𝑋 =
77
7
𝑋 = 11

9. Ungrouped Kurtosis
𝑿𝒊 𝑿𝒊 - 𝑿 (𝑿𝒊 − 𝑿 )
𝟐
(𝑿𝒊 - 𝑿)𝟒
6 6 - 11 -5 25 625
7 7 - 11 -4 16 256
9 9 - 11 -2 4 16
11 11 - 11 0 0 0
13 13 - 11 2 4 16
15 15 - 11 4 16 256
16 16 - 11 5 25 625
= 77 = 90 = 1,794
S =
(𝑿𝒊− 𝑿)𝟐
𝒏−𝟏
S =
90
7−1
S =
90
6
S = 15
S = 𝟑. 𝟖𝟕
v
v

10. Ungrouped Kurtosis
𝑿𝒊 𝑿𝒊 - 𝑿 (𝑿𝒊 − 𝑿 )
𝟐
(𝑿𝒊 - 𝑿)𝟒
6 6 - 11 -5 25 625
7 7 - 11 -4 16 256
9 9 - 11 -2 4 16
11 11 - 11 0 0 0
13 13 - 11 2 4 16
15 15 - 11 4 16 256
16 16 - 11 5 25 625
= 77 = 90 = 1,794
S =
(𝑿𝒊− 𝑿)𝟐
𝒏−𝟏
S =
90
7−1
S =
90
6
S = 15
S = 𝟑. 𝟖𝟕

11. Ungrouped Kurtosis
𝑿𝒊 𝑿𝒊 - 𝑿 (𝑿𝒊 − 𝑿 )
𝟐
(𝑿𝒊 - 𝑿)𝟒
6 6 - 11 -5 25 625
7 7 - 11 -4 16 256
9 9 - 11 -2 4 16
11 11 - 11 0 0 0
13 13 - 11 2 4 16
15 15 - 11 4 16 256
16 16 - 11 5 25 625
= 77 = 90 = 1,794
𝑲 =
(𝑿𝒊 − 𝑿)𝟒
𝒏𝒔𝟒
𝑲 =
1,794
(7)(3.87)4
𝑲 =
1,794
(7)(224.31)
𝑲 =
1,794
1570.17
𝑲 = 1.14
Since K-
computed 1.14
is < 3,
therefore the
distribution is
Platykurtic.

12. Formulas
used in
Grouped data
of Kurtosis

13. For Grouped Data:
Mean:
Standard
Deviation:
𝑿 =
𝑭𝒊𝑿𝒊
𝒏
𝒔 =
𝑭𝒊(𝑿𝒊 − 𝑿)𝟐
𝒏 − 𝟏
𝑲 =
𝑭𝒊(𝑿𝒊 − 𝑿)𝟒
𝒏𝒔𝟒
Where:
Xi = Class Mark
𝑋= Mean
n = Number of
Observations
s = Standard Deviation

14. Grouped
Kurtosis
C.I Frequenc
y
Xi FiXi
76 - 79 2 77.5 155
80 – 83 5 81.5 407.5
84 - 87 5 85.5 427.5
88 - 91 11 89.5 984.5
92 - 95 4 93.5 374
96 - 99 3 97.5 292.5
n = 30 Σ = 2,641
|Xi – X|
10.53
6.53
2.53
1.47
5.47
9.47
𝑿 =
𝑭𝒊𝑿𝒊
𝒏
𝑿 =
2,641
30
𝑿 = 88.03
vv

15. Grouped
Kurtosis
C.I Frequenc
y
Xi FiXi
76 - 79 2 77.5 155
80 – 83 5 81.5 407.5
84 - 87 5 85.5 427.5
88 - 91 11 89.5 984.5
92 - 95 4 93.5 374
96 - 99 3 97.5 292.5
n = 30 Σ = 2,641
|Xi – X|
10.53
6.53
2.53
1.47
5.47
9.47
𝑿 =
𝑭𝒊𝑿𝒊
𝒏
𝑿 =
2,641
30
𝑿 = 88.03

16. C.I Frequenc
y
Xi FiXi |Xi – X| Fi |Xi – X| Fi |Xi - X|² Fi|Xi - X|⁴
76 - 79 2 77.5 155 10.53 21.06 221.7618 24,589.1480
80 – 83 5 81.5 407.5 6.53 32.65 213.2045 9,091.2318
84 - 87 5 85.5 427.5 2.53 12.65 32.0045 204.8576
88 - 91 11 89.5 984.5 1.47 16.17 23.7699 51.3644
92 - 95 4 93.5 374 5.47 21.88 119.6836 3,581.041
96 - 99 3 97.5 292.5 9.47 28.41 269.0427 24,127.9915
n = 30 Σ =
2,641
Σ = 132.82 Σ =
879.467
Σ = 61,645.6343
Grouped
Kurtosis
v
v
vv

17. Grouped
Kurtosis
C.I Frequenc
y
Xi FiXi |Xi – X| Fi |Xi – X| Fi |Xi - X|² Fi|Xi - X|⁴
76 - 79 2 77.5 155 10.53 21.06 221.7618 24,589.1480
80 – 83 5 81.5 407.5 6.53 32.65 213.2045 9,091.2318
84 - 87 5 85.5 427.5 2.53 12.65 32.0045 204.8576
88 - 91 11 89.5 984.5 1.47 16.17 23.7699 51.3644
92 - 95 4 93.5 374 5.47 21.88 119.6836 3,581.041
96 - 99 3 97.5 292.5 9.47 28.41 269.0427 24,127.9915
n = 30 Σ = 2,641 Σ = 132.82 Σ = 879.467 Σ = 61,645.6343

18. Grouped
Kurtosis
𝒔 =
𝑭𝒊(𝑿𝒊 − 𝑿)𝟐
𝒏 − 𝟏
𝒔 =
879.467
30 − 1
𝒔 =
879.467
29
𝒔 = 30.33
𝒔 = 5.51

19. Grouped
Kurtosis
𝑲 =
𝑭𝒊(𝑿𝒊 − 𝑿)𝟒
𝒏𝒔𝟒
𝑲 =
61,645.6343
(30)(5.51)4
𝑲 =
61,645.6343
(30)(921.74)
𝑲 =
61,645.6343
27652.2
𝑲 = 2.23
Since K-
computed 2.23 is <
3, therefore the
distribution is
Platykurtic.

20. Thank you!
Presented by:
Group 2