![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](https://image.slidesharecdn.com/g6statsandprobsreport-220420182943/85/Measures-of-Variation-Ungrouped-Data-15-320.jpg)
![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](https://image.slidesharecdn.com/g6statsandprobsreport-220420182943/85/Measures-of-Variation-Ungrouped-Data-16-320.jpg)
![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](https://image.slidesharecdn.com/g6statsandprobsreport-220420182943/85/Measures-of-Variation-Ungrouped-Data-17-320.jpg)
![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](https://image.slidesharecdn.com/g6statsandprobsreport-220420182943/85/Measures-of-Variation-Ungrouped-Data-18-320.jpg)
![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](https://image.slidesharecdn.com/g6statsandprobsreport-220420182943/85/Measures-of-Variation-Ungrouped-Data-19-320.jpg)
Measures of Variation (Ungrouped Data)
- 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).