Bernard Szlachta: /* Population Variance */

2014-05-27T08:35:37Z

Population Variance

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{{Cat|Summarizing Distributions| 03}}

=What is Variability=
Variability refers to
*how much the numbers in a distribution differ from each other.
*how "spread out" a group of scores is.
The terms variability, spread, and dispersion are synonyms, and refer to how spread out a distribution is.

'''Quiz Example'''

''Quiz1''

[[File:Variability-definition1.jpg|300px]]

''Quiz2''

[[File:Variability-definition.jpg|300px]]

* The graphs above represent the scores on two quizzes.
* The mean score for each quiz is 7.0.
* Despite the equality of means, you can see that the distributions are quite different.
* Specifically, the scores on Quiz 1 are more densely packed and those on Quiz 2 are more spread out.
* The differences among students were much greater on Quiz 2 than on Quiz 1.

=Measures of Variability=
There are four frequently used measures of variability: the range, interquartile range, variance, and standard deviation.

==Range==
* The range is the simplest measure of variability to calculate, and one you have probably encountered many times in your life.
* The range is simply the highest score minus the lowest score.

'''Example 1'''
* What is the range of the following group of numbers: 10, 2, 5, 6, 7, 3, 4?
* The highest number is 10, and the lowest number is 2, so 10 - 2 = 8.
* The range is 8.

'''Example 2'''
* Here’s a dataset with 10 numbers: 99, 45, 23, 67, 45, 91, 82, 78, 62, 51.
* The highest number is 99 and the lowest number is 23, so 99 - 23 equals 76; the range is 76.

'''Quiz Example'''
* Vonsider the two quizzes shown in the graphs above.
* On Quiz 1, the lowest score is 5 and the highest score is 9; the range is 4.
* On Quiz 2, the lowest score is 4 and the highest score is 10; the range is 6.
* The range on Quiz 2 was larger

==Interquartile Range==
* The interquartile range (IQR) is the range of the middle 50% of the scores in a distribution.
* It is computed as follows:
IQR = 75th percentile - 25th percentile

'''Quiz Example'''
* For Quiz 1, the 75th percentile is 8 and the 25th percentile is 6; The interquartile range is 2.
* For Quiz 2, which has greater spread, the 75th percentile is 9, the 25th percentile is 5, and the interquartile range is 4.
* In box plots, the 75th percentile was called the upper hinge and the 25th percentile was called the lower hinge.
* Using this terminology, the interquartile range is referred to as the H-spread.

===semi-interquartile range===
* The semi-interquartile range is defined simply as the interquartile range divided by 2.
* If a distribution is symmetric, the median plus or minus the semi-interquartile range contains half the scores in the distribution.

==Variance==
* Variability can also be defined in terms of how close the scores in the distribution are to the middle of the distribution.
* Using the mean as the measure of the middle of the distribution, the variance is defined as the average squared difference of the scores from the mean.

===Population Variance===
The formula for the variance is:
[[File:Pop_var.gif]]
where σ2 is the variance, μ is the mean, and N is the number of numbers

:[[File:ClipCapIt-140527-093540.PNG]]

'''Quiz Example'''

The data from Quiz 1 are shown in Table below.

{| class="wikitable" style="text-align: center;"
|-
! Scores
! Deviation from Mean
! Squared Deviation
|-
| 9
| 2
| 4
|-
| 9
| 2
| 4
|-
| 9
| 2
| 4
|-
| 8
| 1
| 1
|-
| 8
| 1
| 1
|-
| 8
| 1
| 1
|-
| 8
| 1
| 1
|-
| 7
| 0
| 0
|-
| 7
| 0
| 0
|-
| 7
| 0
| 0
|-
| 7
| 0
| 0
|-
| 7
| 0
| 0
|-
| 6
| -1
| 1
|-
| 6
| -1
| 1
|-
| 6
| -1
| 1
|-
| 6
| -1
| 1
|-
| 6
| -1
| 1
|-
| 5
| -2
| 4
|-
| 5
| -2
| 4
|-
|colspan="3" | Means
|-
| 7
| 0
| 1.5
|}

* The mean score is 7.0.
* Therefore, the column "Deviation from Mean" contains the score minus 7.
* The column "Squared Deviation" is simply the previous column squared.

* The mean deviation from the mean is 0. This will always be the case.
* The mean of the squared deviations is 1.5.
* Therefore, the variance is 1.5.
* Analogous calculations with Quiz 2 show that its variance is 6.7.
* Using the formula, for Quiz 1, μ = 7 and N = 20.

===Sample Variance===
* If the variance in a sample is used to estimate the variance in a population, then the formula for Population Variance underestimates the variance and the following formula should be used is below.
[[File:Sample var.gif]]
where s2 is the estimate of the variance and M is the sample mean.
* Note that M is the mean of a sample taken from a population with a mean of μ.
* Since, in practice, the variance is usually computed in a sample, this formula is most often used.

'''Example'''
* Assume the scores 1, 2, 4, and 5 were sampled from a larger population.
* To estimate the variance in the population you would compute s2 as follows:
M = (1 + 2 + 4 + 5)/4 = 12/4 = 3.
s2 = [(1-3)2 + (2-3)2 + (4-3)2 + (5-3)2]/(4-1)
= (4 + 1 + 1 + 4)/3 = 10/3 = 3.333

===Alternate Formulas ===
* There are alternate formulas that can be easier to use if you are doing your calculations with a hand calculator.
* These formulas are subject to rounding error if your values are very large and/or you have an extremely large number of observations.
[[File:Comp varp.gif]] [[File:Comp var.gif]]

For the example above,

[[File:Formula5.jpg]]

==Standard Deviation==
* The standard deviation is the square root of the variance.

'''Quiz Example'''
The standard deviations of the two quiz distributions 1.225 and 2.588.

===Standard Deviation and Normal Distribution===
* The standard deviation is an especially useful measure of variability when the distribution is normal or approximately normal because the proportion of the distribution within a given number of standard deviations from the mean can be calculated.

* 68% of the distribution is within one standard deviation of the mean and approximately 95% of the distribution is within two standard deviations of the mean.
* Therefore, if you had a normal distribution with a mean of 50 and a standard deviation of 10, then 68% of the distribution would be between 50 - 10 = 40 and 50 +10 =60.

* Similarly, about 95% of the distribution would be between 50 - 2 x 10 = 30 and 50 + 2 x 10 = 70.
* The symbol for the population standard deviation is σ; the symbol for an estimate computed in a sample is s.

'''Example'''

[[File:Std.PNG]]

Figure above shows two normal distributions.
* The red distribution has a mean of 40 and a standard deviation of 5
* The blue distribution has a mean of 60 and a standard deviation of 10
* For the red distribution, 68% of the distribution is between 35 and 45
* for the blue distribution, 68% is between 50 and 70.

==Quiz==
<quiz display=simple >
{ What is the range of 2, 4, 6, and 8?

|type="{}"}
{ 6 }

{
{{Show Answer|
6

8 - 2 is 6
}}
}

{Would the variance of 10, 12, 17, 20, 25, 27, 42, and 45 be larger if the numbers represented a population or a sample?

|type="()"}
-Population
+Sample

{
{{Show Answer|
Sample

The variance would be larger if these numbers represented a sample because you would divide by N-1 (instead of just N).
}}
}

{What is the standard deviation of this sample?
Y
8
15
20
12
13
11
13
15

|type="{}"}
{ 3.5026 }

{
{{Show Answer|
3.5026
}}
}

{What is the interquartile range of these numbers?
Z
12
13
14
15
9
10
16
10
8
10
11
12
13
22
23
24
25

|type="{}"}
{ 9 }

{
{{Show Answer|
9

25th% is 10, 75th% is 19, 19 - 10 is 9
}}
}

Variability - Revision history

Bernard Szlachta: /* Population Variance */