The document discusses normal and skewed distributions. It provides an example of student study hours to illustrate how to create a distribution from a data set. The distribution plots the hours of study on the x-axis and the number of occurrences on the y-axis. It then calculates the mean of the example data set to demonstrate that the mean describes the center point of a normal distribution well.

1. Difference between Normal and
Skewed Distributions

2. This presentation will help you determine if the
data set from the problem you are asked to
solve has a normal or skewed distribution

3. This presentation will help you determine if the
data set from the problem you are asked to
solve has a normal or skewed distribution
Normal
Skewed

4. Knowing if your data’s distribution is skewed or
normal is the second way of knowing if you will
use what is called a parametric or a
nonparametric test

5. The first way (as you may recall from the last
decision point) is to determine if the data is
scaled, ordinal, or nominal

6. But first,

7. What is a distribution?

8. We will illustrate what a distribution is with a
data set that describes the hours students’ study

9. Here is the data set:

10. Student Hours of
Study

11. Student Hours of
Study
Bart 1

12. Student Hours of
Study
Bart 1
Basheba 2

13. Student Hours of
Study
Bart 1
Basheba 2
Bella 2

14. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3

15. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3

16. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3

17. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4

18. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4

19. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5

20. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Data

21. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Data Set

22. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
From this data set we
will create a
distribution:

23. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5

24. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The X Axis, will be the
number of hours of
study

25. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study

26. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Hours of Study
1

27. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2
Hours of Study

28. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3
Hours of Study

29. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4
Hours of Study

30. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study

31. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study

32. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The Y Axis, indicates
the number of times
the same number
occurs
1 2 3 4 5
Hours of Study

33. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The Y Axis, indicates
the number of times
the same number
occurs
1 2 3 4 5
Hours of Study

34. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The Y Axis, indicates
the number of times
the same number
occurs
1 2 3 4 5
Hours of Study

35. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The Y Axis, indicates
the number of times
the same number
occurs
1 2 3 4 5
Hours of Study

36. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The Y Axis, indicates
the number of times
the same number
occurs
1 2 3 4 5
Hours of Study

37. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
The Y Axis, indicates
the number of times
the same number
occurs
1 2 3 4 5
Hours of Study

38. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
Number of Occurrences
1 2 3 4 5
Hours of Study

39. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences

40. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences

41. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

42. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

43. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

44. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

45. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

46. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

47. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

48. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

49. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

50. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

51. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

52. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

53. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

54. Student Hours of
Study
Bart 1
Basheba 2
Bella 2
Bob 3
Boston 3
Bunter 3
Buxby 4
Bybee 4
Bwinda 5
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1
This is a
distribution

55. One way to represent a distribution like this:

56. One way to represent a distribution like this:

57. One way to represent a distribution like this:
Is like this:

58. One way to represent a distribution like this:
Is like this:

59. One way to represent a distribution like this:
Is like this:
Normal distributions have
the majority of the data in
the middle

60. One way to represent a distribution like this:
Is like this:
Normal distributions have
the majority of the data in
the middle

61. One way to represent a distribution like this:
Is like this:
With decreasing
but equal amounts
toward the tails

62. One way to represent a distribution like this:
Is like this:
With decreasing
but equal amounts
toward the tails
With decreasing
but equal amounts
toward the tails

63. The mean or average works really well with
normal distributions

64. Another way to say it, is that the mean describes
well the center point of a normal distribution

66. A Normal Distribution

67. The Mean

68. Here is how you calculate the mean:

70. Let’s put the data into the
distribution

71. 2
1 2
3
3
3
4
4
5

72. 2
1 2
3
3
3
4
4
5
Mean =

73. 2
1 2
3
3
3
4
4
5
Mean =

74. 2
1 2
3
3
3
4
4
5
Mean =

75. 2
1 2
3
3
3
4
4
5
Mean =
ퟏ

76. 2
1 2
3
3
3
4
4
5
Mean =
1+ퟐ

77. 2
1 2
3
3
3
4
4
5
Mean =
1+2+ퟐ

78. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+ퟑ

79. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+ퟑ

80. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+ퟑ

81. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+ퟒ

82. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+ퟒ

83. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+4+ퟓ

84. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+4+5
Divided by the number of
total values

85. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+4+5
Divided by the number of
total values

86. Mean =
1+2+2+3+3+3+4+4+5
ퟗ
2
1 2
3
3
3
4
4
5

87. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+4+5
9
=
27
9

88. 2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+4+5
9
=
27
9
= 3

89. 2
1 2
3
3
3
4
4
5
Mean = 3

90. 2
1 2
3
3
3
4
4
5
Mean = 3

91. The mean is a good estimate of the center of a
distribution when the distribution is normal

92. But, the mean is not a good estimate of the
center when the distribution is not normal

93. This is because of what we call OUTLIERS

94. What is an outlier?

95. An outlier is a data point that falls outside the
overall pattern of the distribution

96. As an example, here is the overall pattern

97. As an example, here is the overall pattern
2
1 2
3
3
3
4
4
5

98. But what if we changed the 5

99. But what if we changed the 5
2
1 2
3
3
3
4
4
5

100. to a 50

101. to a 50
2
1 2
3
3
3
4
4
50

102. to a 50
2
1 2
3
3
3
4
4
50

103. To illustrate what happens to the mean when an
outlier is present, let’s go back to this
distribution:

104. To illustrate what happens to the mean when an
outlier is present, let’s go back to this
distribution:
2
1 2
3
3
3
4
4
5

105. Let’s say one student, instead of studying five
hours studies 23 hours a day!!!!!

106. Watch what happens to the mean:

107. Before

108. Mean =
1+2+2+3+3+3+4+4+5
9
=
27
9
= 3
2
1 2
3
3
3
4
4
5

109. After

110. 2
1 2
3
3
3
4
4
5

111. 2
1 2
3
3
3
4
4
23

112. 2
1 2
3
3
3
4
4
23

113. 2
1 2
3
3
3
4
4
23
Mean =
1+2+2+3+3+3+4+4+ퟐퟑ
9
=

114. 2
1 2
3
3
3
4
4
23
Mean =
1+2+2+3+3+3+4+4+23
9
=
ퟒퟓ
ퟗ

115. 2
1 2
3
3
3
4
4
23
Mean =
1+2+2+3+3+3+4+4+23
9
=
45
9
= ퟓ

116. Once again, BEFORE

117. Once again, BEFORE
2
1 2
3
3
3
4
4
5
Mean =
1+2+2+3+3+3+4+4+5
9
=
27
9
= ퟑ

118. AFTER
2
1 2
3
3
3
4
4
23
Mean =
1+2+2+3+3+3+4+4+23
9
=
45
9
= ퟓ

119. Just by changing one value from “5” to “23” the
mean changed by two values (from “3” to “5”)

120. Thus, the mean is very sensitive to outliers

121. Therefore, the mean is not a good estimate of
the center of a distribution when the
distribution is NOT NORMAL

122. Therefore, the mean is not a good estimate of
the center of a distribution when the
distribution is NOT NORMAL

123. Therefore, the mean is not a good estimate of
the center of a distribution when the
distribution is NOT NORMAL

124. Therefore, the mean is not a good estimate of
the center of a distribution when the
distribution is NOT NORMAL

125. Here is a guiding principle

126. 1 If your data set is normally distributed like
this:

127. 1 If your data set is normally distributed like this:
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

128. 1 If your data set is normally distributed like this,
then you will use a parametric test

129. 2

130. 2 If your data set is skewed either to the right

131. 2 If your data set is skewed either to the right
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

132. 2 If your data set is skewed either to the right
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

133. 2 If your data set is skewed either to the right
or to the left

134. 2 If your data set is skewed either to the right
or to the left
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

135. 2 If your data set is skewed either to the right
or to the left
1 2 3 4 5
Hours of Study
Number of Occurrences
3
2
1

136. 2 If your data set is skewed either to the right
or to the left, then you will use a nonparametric
test

137. In summary,

138. In summary,
A parametric test is used when the problem’s
data set is normally distributed

139. In summary,
A parametric test is used when the problem’s
data set is normally distributed

140. In summary,
A parametric test is used when the problem’s
data set is normally distributed
A non-parametric test is used when the
problem’s data set is very skewed to the right or
the left:

141. In summary,
A parametric test is used when the problem’s
data set is normally distributed
A non-parametric test is used when the
problem’s data set is very skewed to the right or
the left:

142. In summary,
A parametric test is used when the problem’s
data set is normally distributed
A non-parametric test is used when the
problem’s data set is very skewed to the right or
the left:

143. In summary,
A parametric test is used when the problem’s
data set is normally distributed:
A non-parametric test is used when the
problem’s data set is very skewed to the right or
the left:
Or very non-normal:

144. In summary,
A parametric test is used when the problem’s
data set is normally distributed:
A non-parametric test is used when the
problem’s data set is very skewed to the right or
the left:
Or very non-normal:

145. So, how do you know if your data is normally
distributed?

146. So, how do you know if your data is normally
distributed?
Go to the Learning Module entitled: Assessing
Skew. You will find it next to the link for this
presentation.

147. So, how do you know if your data is normally
distributed?
Go to the Learning Module entitled: Assessing
Skew. You will find it next to the link for this
presentation.
After you have viewed that learning module use
SPSS to assess the skew of your data.

148. Is your data normally distributed or skewed?

149. If your data was skewed with a critical ratio
greater than 2.0 or less than -2.0 then select
Skewed

150. Otherwise select
Normal

151. It is important to note that if you choose
Skewed, your data will be analyzed using what
are called non-parametric tests
Skewed

152. Non-parametric tests differ from parametric
tests in one simple way:

153. Parametric tests use the mean in their
calculations

154. Parametric tests use the mean in their
calculations
Non-parametric tests use the median

155. What is the median?

156. The median is simply the middle score of a data
set where

157. The median is simply the middle score of a data
set where
• 50% of the scores fall below it and

158. The median is simply the middle score of a data
set where
• 50% of the scores fall below it and
• 50% of the scores are above it

159. To illustrate let’s go back to this distribution:

160. To illustrate let’s go back to this distribution:
2
1 2
3
3
3
4
4
5

161. With the Median we simply determine the mid
point:
2
1 2
3
3
3
4
4
5

162. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
5

163. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
5
4 units

164. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
5

165. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
5

166. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
5

167. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
5

168. Notice that the Median is unaffected by outliers

169. To illustrate this, we’ll change the value “5” to a
“10”:

170. 2
1 2
3
3
3
4
4
5

171. 2
1 2
3
3
3
4
4
10

172. Watch what happens to the median:
2
1 2
3
3
3
4
4
10

173. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +10
2
1 2
3
3
3
4
4
10

174. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +10
2
1 2
3
3
3
4
4
10
4 units

175. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +10
2
1 2
3
3
3
4
4
10
4 units
4 units

176. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +10
2
1 2
3
3
3
4
4
10
4 units
4 units

177. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5
2
1 2
3
3
3
4
4
10
Hmm, it’s
still 3

178. But, what if we change the value 10 to 1,000!!!

179. Watch again what happens to the median:
2
1 2
3
3
3
4
4
1,000

180. Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +1000
2
1 2
3
3
3
4
4
1,000

181. 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +1000
2
1 2
3
3
3
4
4
1,000

182. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +1000
2
1 2
3
3
3
4
4
1,000

183. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 +1000
2
1 2
3
3
3
4
4
1,000

184. 4 units 4 units
Median = 1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 1000
2
1 2
3
3
3
4
4
What do you
know –
It’s still 3
1,000

185. Here is the key take away:

186. The mean is affected by outliers

187. The mean is affected by outliers
The median is not affected by outliers

188. Therefore the mean is used with more or less
NORMAL DISTRIBUTIONS

189. Therefore the mean is used with more or less
NORMAL DISTRIBUTIONS

190. And the median is used with SKEWED OR NON-NORMAL
DISTRIBUTIONS

191. And the median is used with SKEWED OR NON-NORMAL
DISTRIBUTIONS

192. And the median is used with SKEWED OR NON-NORMAL
DISTRIBUTIONS

193. And the median is used with SKEWED OR NON-NORMAL
DISTRIBUTIONS

194. So, why doesn’t everyone use non-parametric
methods since they are unaffected by outliers?

195. Because parametric methods provide more
meaningful information about the population
than do non-parametric methods

196. So, if your data is skewed it’s better to get what
information you can from a non-parametric
test,

197. So, if your data is skewed it’s better to get what
information you can from a non-parametric
test, even though a parametric test would have
provided more information (if your data had
been normally distributed)

198. So, based on your analysis, which distribution
best reflect your data set:

199. So, based on your analysis, which distribution
best reflect your data set:
Normal
Skewed