Ahnboyoung: /* Skewness */

2014-05-31T21:09:26Z

Skewness

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[[Category:Introduction to Statistics| 11]]
=Discrete Variable=
==Frequency tables==
* containing the number of occurrences in each class of data
* often used to create histograms and frequency polygons
{| class="wikitable" style="text-align: center;"
|-
! Colour
! Frequency
| rowspan="7"| [[File:ClipCapIt-140530-234847.PNG|230px]]
|-
| Brown
| 17
|-
| Yellow
| 18
|-
| Red
| 7
|-
| Green
| 7
|-
| Blue
| 2
|-
| Orange
| 4
|-
|}

==Frequency Distribution==
* the distribution of empirical data
* consists of a count of the number of occurrences of each value
* For a discrete random variable, a probability distribution contains the probability of each possible outcome
* The sum of all probabilities is always 1.0

{| class="wikitable" style="text-align: center;"
|-
! Frequency Distribution
! Probability Distribution
|-
| [[File:ClipCapIt-140530-234950.PNG|300px]]
| [[File:ClipCapIt-140530-235232.PNG|300px]]
|-
|}

=Continuous Variable Distribution=
Problems?
Response time (in millisecond)
568 577 581 640 641 645 657
673 696 703 720 728 729 777
808 824 825 865 875 1007

==Grouped Frequency Distribution==
* a frequency distribution in which frequencies are displayed for ranges of data rather than for individual values.
* '''Histogram''' is a graphical representation of a distribution .
* It partitions the variable on the x-axis into various contiguous class intervals of (usually) equal widths.

;Example
{| class="wikitable" style="text-align: center;"
|-
! Range
! Frequency
| rowspan="7"| [[File:ClipCapIt-140531-000203.PNG|300px]]
|-
| 500-600
| 3
|-
| 600-700
| 6
|-
| 700-800
| 5
|-
| 800-900
| 5
|-
| 900-1000
| 0
|-
| 1000-1100
| 1
|-
|}

=A probability density function=
A probability density function is a formula that can be used to compute probabilities of a range of outcomes for a continuous random variable.

==Normal Distribution==
* one of the most common continuous distributions
* sometimes referred to as a "bell-shaped distribution.
:[[File:ClipCapIt-140531-000739.PNG]]

==Skewness==
A distribution is skewed if one tail extends out further than the other.
* A distribution has positive skew (is skewed to the right) if the tail to the right is longer
* A distribution has a negative skew (is skewed to the left) if the tail to the left is longer

:[[File:ClipCapIt-140531-220924.PNG]]

==Kurtosis==
* Leptokurtic is a distribution with long tails relative to a normal distribution
* Platykurtic is a distribution with short tails relative to a normal distribution
:[[File:ClipCapIt-140531-001710.PNG]]

=Quiz=
<quiz display=simple >

{ A frequency distribution contains the frequency of every value in the distribution.

|type="()"}
+True
-False

{
{{Show Answer|
True

The distribution of empirical data is called a frequency distribution and consists of a count of the number of occurrences of each value.
}}
}

{ A grouped frequency distribution should be used instead of a frequency distribution when the

|type="()"}
-distribution is bimodal.
-distribution is skewed.
+variable is continuous.

{
{{Show Answer|
variable is continuous.

When a variable is truly continuous, each value will have a frequency of 1. Therefore, grouped frequency distributions are needed with continuous variables. }}
}

{ A symmetric distribution

|type="()"}
-has equal positive and negative skews.
+has no skew.
-can have either positive or negative skew, but not both.

{
{{Show Answer|
has no skew.

In a symmetric distribution, the tails extend equally in both directions. Therefore, there is no skew.
}}
}

{ The following distribution has
:[[File:ClipCapIt-140531-002415.PNG]]

|type="()"}
+a positive skew.
-a negative skew.
-no skew.

{
{{Show Answer|
a positive skew.

The tail in the positive direction is longer than the tail in the negative direction, thus it has a positive skew.
}}
}

{ The area under the curve of a probability distribution is

|type="{}"}
{ 1 }

{
{{Show Answer|
1

The area is 1 by definition, meaning that the probability that a score chosen at random will occur under the curve is 1.
}}
}

{ A normal or bell-shaped distribution has its greatest probability density in its tails.

|type="()"}
-True
+False

{
{{Show Answer|
False

The distribution is higher and therefore denser in the middle of the distribution.
}}
}

{ Which of the following distributions is/are symmetric?

:[[File:ClipCapIt-140531-002800.PNG]]

|type="[]"}
+A
-B
-C
+D

{
{{Show Answer|
A, D

A and D are symmetric, meaning if you folded them in the middle, the two sides would match perfectly. Distributions B and C have positive skew.}}
}

</quiz>

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Ahnboyoung: /* Skewness */