Difference between revisions of "Calculate Standard Deviation"
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| − | The standard deviation calculation tells you how spread out the numbers are in your sample.<ref>http://www.mathsisfun.com/data/standard-deviation.html</ref> This | + | The standard deviation calculation tells you how spread out the numbers are in your sample.<ref name="rf6472">http://www.mathsisfun.com/data/standard-deviation.html</ref> This article will show you how to find the mean, variance, and standard deviation. |
== 10 Second Summary == | == 10 Second Summary == | ||
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4. Divide the sum by (n-1) to get the variance.<br> | 4. Divide the sum by (n-1) to get the variance.<br> | ||
5. Take the square root of the variance, then check your work. | 5. Take the square root of the variance, then check your work. | ||
| − | [[Category:Probability and Statistics]] | + | [[Category: Probability and Statistics]] |
==Steps== | ==Steps== | ||
===Finding the Mean=== | ===Finding the Mean=== | ||
| − | #Look at your data set. This is an important step in any type of statistical calculation, even if it is a simple figure like the mean or median.<ref | + | #Look at your data set. This is an important step in any type of statistical calculation, even if it is a simple figure like the mean or median.<ref name="rf6472" /> |
#*Know how many numbers are in your sample. | #*Know how many numbers are in your sample. | ||
#*Do the numbers vary across a large range? Or are the differences between the numbers small, such as just a few decimal places? | #*Do the numbers vary across a large range? Or are the differences between the numbers small, such as just a few decimal places? | ||
#*Know what type of data you are looking at. What do your numbers in your sample represent? this could be something like test scores, heart rate readings, height, weight etc. | #*Know what type of data you are looking at. What do your numbers in your sample represent? this could be something like test scores, heart rate readings, height, weight etc. | ||
#*For example, a set of test scores is 10, 8, 10, 8, 8, and 4. | #*For example, a set of test scores is 10, 8, 10, 8, 8, and 4. | ||
| − | #Gather all of your data. You will need every number in your sample to calculate the mean.<ref | + | #Gather all of your data. You will need every number in your sample to calculate the mean.<ref name="rf6472" /> |
#*The mean is the average of all your data points. | #*The mean is the average of all your data points. | ||
#*This is calculated by adding all of the numbers in your sample, then dividing this figure by the how many numbers there are in your sample (n). | #*This is calculated by adding all of the numbers in your sample, then dividing this figure by the how many numbers there are in your sample (n). | ||
#*In the sample of test scores (10, 8, 10, 8, 8, 4) there are 6 numbers in the sample. Therefore n = 6. | #*In the sample of test scores (10, 8, 10, 8, 8, 4) there are 6 numbers in the sample. Therefore n = 6. | ||
| − | #Add the numbers in your sample together. This is the first part of calculating a mathematical average or mean.<ref | + | #Add the numbers in your sample together. This is the first part of calculating a mathematical average or mean.<ref name="rf6472" /> |
#*For example, use the data set of quiz scores: 10, 8, 10, 8, 8, and 4. | #*For example, use the data set of quiz scores: 10, 8, 10, 8, 8, and 4. | ||
#*10 + 8 + 10 + 8 + 8 + 4 = 48. This is the sum of all the numbers in the data set or sample. | #*10 + 8 + 10 + 8 + 8 + 4 = 48. This is the sum of all the numbers in the data set or sample. | ||
#*Add the numbers a second time to check your answer. | #*Add the numbers a second time to check your answer. | ||
| − | #Divide the sum by how many numbers there are in your sample (''n''). This will provide the average or mean of the data.<ref | + | #Divide the sum by how many numbers there are in your sample (''n''). This will provide the average or mean of the data.<ref name="rf6472" /> |
#*In the sample of test scores (10, 8, 10, 8, 8, and 4) there are six numbers, so n = 6. | #*In the sample of test scores (10, 8, 10, 8, 8, and 4) there are six numbers, so n = 6. | ||
#*The sum of the test scores in the example was 48. So you would divide 48 by n to figure out the mean. | #*The sum of the test scores in the example was 48. So you would divide 48 by n to figure out the mean. | ||
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#*The mean test score in the sample is 8. | #*The mean test score in the sample is 8. | ||
===Finding the Variance In Your Sample=== | ===Finding the Variance In Your Sample=== | ||
| − | #Find the variance. The variance is a figure that represents how far the data in your sample is clustered around the mean.<ref>http://pirate.shu.edu/~wachsmut/Teaching/MATH1101/Descriptives/variability.html</ref> | + | #Find the variance. The variance is a figure that represents how far the data in your sample is clustered around the mean.<ref name="rf6473">http://pirate.shu.edu/~wachsmut/Teaching/MATH1101/Descriptives/variability.html</ref> |
#*This figure will give you an idea of how far your data is spread out. | #*This figure will give you an idea of how far your data is spread out. | ||
#*Samples with low variance have data that is clustered closely about the mean. | #*Samples with low variance have data that is clustered closely about the mean. | ||
#*Samples with high variance have data that is clustered far from the mean. | #*Samples with high variance have data that is clustered far from the mean. | ||
#*Variance is often used to compare the distribution of two data sets. | #*Variance is often used to compare the distribution of two data sets. | ||
| − | #Subtract the mean from each of your numbers in your sample. This will give you a figure of how much each data point differs from the mean.<ref | + | #Subtract the mean from each of your numbers in your sample. This will give you a figure of how much each data point differs from the mean.<ref name="rf6473" /> |
#*For example, in our sample of test scores (10, 8, 10, 8, 8, and 4) the mean or mathematical average was 8. | #*For example, in our sample of test scores (10, 8, 10, 8, 8, and 4) the mean or mathematical average was 8. | ||
#* 10 - 8 = 2; 8 - 8 = 0, 10 - 8 = 2, 8 - 8 = 0, 8 - 8 = 0, and 4 - 8 = -4. | #* 10 - 8 = 2; 8 - 8 = 0, 10 - 8 = 2, 8 - 8 = 0, 8 - 8 = 0, and 4 - 8 = -4. | ||
#*Do this procedure again to check each answer. It is very important you have each of these figures correct as you will need them for the next step. | #*Do this procedure again to check each answer. It is very important you have each of these figures correct as you will need them for the next step. | ||
| − | #Square all of the numbers from each of the subtractions you just did. You will need each of these figures to find out the variance in your sample.<ref | + | #Square all of the numbers from each of the subtractions you just did. You will need each of these figures to find out the variance in your sample.<ref name="rf6473" /> |
#*Remember, in our sample we subtracted the mean (8) from each of the numbers in the sample (10, 8, 10, 8, 8, and 4) and came up with the following: 2, 0, 2, 0, 0 and -4. | #*Remember, in our sample we subtracted the mean (8) from each of the numbers in the sample (10, 8, 10, 8, 8, and 4) and came up with the following: 2, 0, 2, 0, 0 and -4. | ||
#*To do the next calculation in figuring out variance you would perform the following: 2<sup>2</sup>, 0<sup>2</sup>, 2<sup>2</sup>, 0<sup>2</sup>, 0<sup>2</sup>, and (-4)<sup>2</sup> = 4, 0, 4, 0, 0, and 16. | #*To do the next calculation in figuring out variance you would perform the following: 2<sup>2</sup>, 0<sup>2</sup>, 2<sup>2</sup>, 0<sup>2</sup>, 0<sup>2</sup>, and (-4)<sup>2</sup> = 4, 0, 4, 0, 0, and 16. | ||
#*Check your answers before proceeding to the next step. | #*Check your answers before proceeding to the next step. | ||
| − | #Add the squared numbers together. This figure is called the sum of squares.<ref | + | #Add the squared numbers together. This figure is called the sum of squares.<ref name="rf6473" /> |
#*In our example of test scores, the squares were as follows: 4, 0, 4, 0, 0, and 16. | #*In our example of test scores, the squares were as follows: 4, 0, 4, 0, 0, and 16. | ||
#*Remember, in the example of test scores we started by subtracting the mean from each of the scores and squaring these figures: (10-8)^2 + (8-8)^2 + (10-8)^2 + (8-8)^2 + (8-8)^2 + (4-8)^2 | #*Remember, in the example of test scores we started by subtracting the mean from each of the scores and squaring these figures: (10-8)^2 + (8-8)^2 + (10-8)^2 + (8-8)^2 + (8-8)^2 + (4-8)^2 | ||
#*4 + 0 + 4 + 0 + 0 + 16 = 24. | #*4 + 0 + 4 + 0 + 0 + 16 = 24. | ||
#*The sum of squares is 24. | #*The sum of squares is 24. | ||
| − | #Divide the sum of squares by (n-1). Remember, n is how many numbers are in your sample. Doing this step will provide the variance. The reason to use n-1 is to have sample variance and population variance unbiased. <ref | + | #Divide the sum of squares by (n-1). Remember, n is how many numbers are in your sample. Doing this step will provide the variance. The reason to use n-1 is to have sample variance and population variance unbiased. <ref name="rf6473" /> |
#*In our sample of test scores (10, 8, 10, 8, 8, and 4) there are 6 numbers. Therefore, n = 6. | #*In our sample of test scores (10, 8, 10, 8, 8, and 4) there are 6 numbers. Therefore, n = 6. | ||
#*n-1 = 5. | #*n-1 = 5. | ||
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#*The variance in this sample is thus 4.8. | #*The variance in this sample is thus 4.8. | ||
===Calculating the Standard Deviation=== | ===Calculating the Standard Deviation=== | ||
| − | #Find your variance figure. You will need this to find the standard deviation for your sample.<ref | + | #Find your variance figure. You will need this to find the standard deviation for your sample.<ref name="rf6473" /> |
#*Remember, variance is how spread out your data is from the mean or mathematical average. | #*Remember, variance is how spread out your data is from the mean or mathematical average. | ||
#*Standard deviation is a similar figure, which represents how spread out your data is in your sample. | #*Standard deviation is a similar figure, which represents how spread out your data is in your sample. | ||
#*In our example sample of test scores, the variance was 4.8. | #*In our example sample of test scores, the variance was 4.8. | ||
| − | #Take the square root of the variance. This figure is the standard deviation.<ref | + | #Take the square root of the variance. This figure is the standard deviation.<ref name="rf6473" /> |
#*Usually, at least 68% of all the samples will fall inside one standard deviation from the mean. | #*Usually, at least 68% of all the samples will fall inside one standard deviation from the mean. | ||
#*Remember in our sample of test scores, the variance was 4.8. | #*Remember in our sample of test scores, the variance was 4.8. | ||
#* √4.8 = 2.19. The standard deviation in our sample of test scores is therefore 2.19. | #* √4.8 = 2.19. The standard deviation in our sample of test scores is therefore 2.19. | ||
#*5 out of 6 (83%) of our sample of test scores (10, 8, 10, 8, 8, and 4) is within one standard deviation (2.19) from the mean (8). | #*5 out of 6 (83%) of our sample of test scores (10, 8, 10, 8, 8, and 4) is within one standard deviation (2.19) from the mean (8). | ||
| − | #Go through finding the mean, variance and standard deviation again. This will allow you to check your answer.<ref | + | #Go through finding the mean, variance and standard deviation again. This will allow you to check your answer.<ref name="rf6473" /> |
#*It is important that you write down all steps to your problem when you are doing calculations by hand or with a calculator. | #*It is important that you write down all steps to your problem when you are doing calculations by hand or with a calculator. | ||
#*If you come up with a different figure the second time around, check your work. | #*If you come up with a different figure the second time around, check your work. | ||
#*If you cannot find where you made a mistake, start over a third time to compare your work. | #*If you cannot find where you made a mistake, start over a third time to compare your work. | ||
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== Related Articles == | == Related Articles == | ||
