Measures of Variation: Range Variance and Standard Deviation (Ungrouped Data) with quiz at the last three pages.

1. Measures of
Variation: Range
Variance and
Standard Deviation

2. Variation is a way to
show how data is
dispersed, or spread out.
Several measures of variation
are used in statistics.
Introduction

3. Measures of
Variation:
Range Variance
Definition and Examples

4. Example 01
Find and interpret the range in length of
Burmese pythons.
Lengths (ft)
18.5
11
14
12.5
16.25
8
10
15.5
6.25
5
Range is one of
the ways to know
how spread out a
certain data is.
First, it needs to be organized with an increasing order.
5, 6.25, 8, 10, 11, 12.5, 14, 15.5, 16.25, 18. 5
18.5 is the greatest value
18.5
Meanwhile, 5 is the least value.
5
Range = difference between the greatest value and the least value.
So it will be 18.5 minus 5, that results with 13. 5.
So the range is 13.5

5. Example 02
Let’s make it simpler, find the range with
the given set below.
3, 8, 10, 3, 2, 5, 7, 9, 9 Again, let’s
arrange it with
an increasing
order.
Result: 2, 3, 3, 5, 7, 8, 9, 9, 10
Highest value is 10
And the lowest value is 2
So, 10 – 2 = 8
Therefore, 8 is our range.

6. Example 03
Find the interquartile range of the following
data set.
What is an
interquartile range?
As the word depicts,
quartile divides the data
set in 4 equal groups.
18, 21, 22, 24, 28, 30, 31, 32, 36, 37
1. Find the median. Medians mean middle. And because, there are two numbers, 28 and
30. The median is obviously 29.
2. The 5 digits before the median are considered as the lower half. And the lower half
have the 1st quartile or lower quartile, 22, since it is the number in the middle.
3. Meanwhile, the 5 digits after the median are considered as the upper half. And this
upper half have the 3rd quartile or upper quartile, 32.
4. To find the interquartile range or IQR, we must find the difference between the 1st
quartile from the 3rd quartile. So 32 minus 22 is 10. There it is, the interquartile
range is 10.
Lower half Upper half

7. Example 04
Find and interpret the interquartile range of
the data.
220, 230, 230, 240, 240, 245, 250, 250, 250, 260, 260, 270
Find the median. The median is found between 245 and 250 so it
will be 247.5
Median = 247.5
1st quartile = 235 3rd quartile = 255
IQR = 255 – 235
= 20
The lower or first quartile is found between 230 and 240 so it will be 235.
Meanwhile, the upper or third quartile is found between 250 and 260 so it will be 255.
And to find the interquartile range, 255 minus 235 will be 20. That’s it, the interquartile range is 20.

8. Example 05
Check for outliers from Example 03.
220, 230, 230, 240, 240, 245, 250, 250, 250, 260, 260, 270
Median = 247.5
1st quartile = 235 3rd quartile = 255
IQR = 20
Outliers is same as checking.
The formula that will be used to check outliers is:
1st Quartile (Q1) – 1.5 (IQR)
If your graph shows a number lower than the
outcome, then there are outliers.
= Q1 – 1.5 (IQR)
= 235 – 1.5 (20)
= 235 – 30
= 205
= Q3 + 1.5 (IQR)
= 255 + 1.5 (20)
= 255 + 30
= 285
There is no data
value lower than 205
in the graph and
higher than 285.
So the graph
consists no
outliers.
And
3rd quartile (Q3) + 1.5 (IQR)
If your graph shows a number higher than the
outcome, then there are outliers.

9. Definition and Examples
Measures of
Variation: Standard
Deviation

10. The standard deviation is a statistic that
measures the dispersion of a dataset
relative to its mean and is calculated as
the square root of the variance.
Sample Standard
Deviation

11. Sample Standard Deviation
Calculate the standard deviation of the
following set of numbers:
82, 93, 98, 89, and 88
First, calculate the mean.
Mean =
sum
n
=
82 + 93 + 98 + 89 + 88
5
= 450
5
= 90
Now, for the standard deviation.
The formula that will be used is…
Sample mean
The n that can be seen
below in the formula in
finding the mean
defines the number of
value seen in the set of
numbers.

12. √
Sample Standard Deviation
(82-90) 2 + (93-90)2 + (98-90) 2 + (89-90) 2 + (88-90) 2
5 – 1
√
(-8) 2 + (3)2 + (8) 2 + (-1) 2 + (-2) 2
4
5.958
√
64 + 9 + 64 + 1 + 4
4
=
142
4
=
=
=
=
=
√35.5
√

13. Sample Standard Deviation
Let’s make it easier with a certain technique
that’ll probably make you understand
more.
Find the standard deviation
of scores 6, 7, 8, 9, 10
6
7
8
9
10
Total = 10 Variance = 10/5 = 2
Standard Deviation: √2 = 1.41
Total = 40
Mean = 40/5 = 8
-2
-1
0
1
2
8
8
8
8
8
4
1
0
1
4

14. Population standard deviation looks at the
square root of the variance of the set of
numbers. It's used to determine a
confidence interval for drawing conclusions
(such as accepting or rejecting
a hypothesis).
Population Standard
Deviation

15. Population Standard
Deviation
There are different ways to write out the steps of the
population standard deviation calculation into an equation.
A common equation is:
σ = ([Σ(x - u)2]/N)1/2
Where:
σ is the population standard deviation
Σ represents the sum or total from 1 to N
x is an individual value
u is the average of the population
N is the total number of the population

16. Population Standard
Deviation
σ = ([Σ(x - u)2]/N)1/2
Example Problem
You grow 20 crystals from a solution and measure the length of each crystal
in millimeters. Here is your data:
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4
Calculate the population standard deviation of the length of the crystals.
First Step. Calculate the mean of the data. Add up all the numbers and divide
by the total number of data points.
So the mean is 7
9+2+5+4+12+7+8+11+9+3+7+4+12+5+4+10+9+6+9+4
20
=
140
20
= 7

17. Population Standard
Deviation
σ = ([Σ(x - u)2]/N)1/2
Second step. Subtract the mean from each data point (or the other
way around, if you prefer, you will be squaring this number, so it does
not matter if it is positive or negative).
(9 - 7)2 = (2)2 = 4
(2 - 7)2 = (-5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(7 - 7)2 = (0)2 = 0
(8 - 7)2 = (1)2 = 1
(11 - 7)2 = (4)22 = 16
(9 - 7)2 = (2)2 = 4
(3 - 7)2 = (-4)22 = 16
(7 - 7)2 = (0)2 = 0
(4 - 7)2 = (-3)2 = 9
(12 - 7)2 = (5)2 = 25
(5 - 7)2 = (-2)2 = 4
(4 - 7)2 = (-3)2 = 9
(10 - 7)2 = (3)2 = 9
(9 - 7)2 = (2)2 = 4
(6 - 7)2 = (-1)2 = 1
(9 - 7)2 = (2)2 = 4
(4 - 7)2 = (-3)22 = 9

18. Population Standard
Deviation
σ = ([Σ(x - u)2]/N)1/2
Third Step. Calculate the mean of the squared
differences.
4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1+4+9
20
=
178
20
= 8.9
The variance is 8.9

19. Population Standard
Deviation
σ = ([Σ(x - u)2]/N)1/2
Lastly. The population standard deviation
is the square root of the variance.
Use a calculator to obtain this number.
(8.9)1/2 = 2.983
The population standard deviation is 2.983

20. “The population standard deviation is a
parameter, which is a fixed value
calculated from every individual in the
population. A sample standard deviation
is a statistic. This means that it is
calculated from only some of the
individuals in a population.”

21. Quiz
Let’s apply what we’ve
learned earlier with the
questions provided by
the reporters, goodluck!

22. Quiz
1. Find and interpret the range of
the following set of numbers:
3, 5, 9, 12, 17, 18, 20
2. With the example of number
one (1), what is the median?

23. Quiz
The ages of people in line for a
roller coaster are 15, 17, 21, 32,
41,30, 25, 52, 16, 39, 11, and 24.
3. Find the range.
4. Find the interquartile range.
5. Define the median?
6. Define the first quartile?
7. The third quartile?

24. Quiz
8. In the data set 9, 6, 8,5, 7, find
the sample standard deviation.
9. Find the population standard
deviation from the following data
set: 2, 4, 4, 4, 5, 5, 7, 9.
10. Define the mean in number
nine (9).