A step-by-step walkthrough Standard deviation A step-by-step walkthrough

Set up Take a set of numbers, any set of numbers, it doesn’t matter if they are inches or kilometers, ounces or ages, weights or heights, as long as they are all heights, or all ages, or all basketball scores. It also does not matter what order they are arranged in, the just have to be numbers that are in some way or another related to each other. In this case, we will use the numbers: 22, 15, 35, 24, 15, 40, 49, 20, 28, 38. We can denote this as a set: Set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 }. Each number in the set is an observation. Since there are ten numbers, there are ten observations.

Standard deviation: step one: finding the mean The MEAN is a mathematical term synonymous with arithmetic average. To take a mean, we add up all of the numbers and divide by the number of observation. In this case, we will use the same set. Set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 }. First, we add the numbers together: 22 + 15 + 35 + 24 + 15 + 40 + 49 + 20 + 28 + 38 = 286 Then, we divide by the number of observations, which we have already determined is ten. 286 = 28.6 10 The MEAN of the set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } is 28.6.

Standard deviation: step two: calculating deviations Let’s order the set vertically to make it easier: The deviation is the difference between the observation and the mean. To find the deviations, subtract the mean from each observation: observation – mean = deviation. Observations Mean Deviation 22 – = 15 – = 35 – = 24 – = 15 – = Try it before I give you the answers. Then match your answers with mine on the next two slides. 40 – = 49 – = 20 – = 28 – = 38 – =

Standard deviation: step two: calculating deviations Let’s order the set vertically to make it easier: The deviation is the difference between the observation and the mean. To find the deviations, subtract the mean from each observation: observation – mean = deviation. Observations Mean Deviation 22 28.6 – = 15 28.6 – = 35 28.6 – = 24 28.6 – = 15 28.6 – = Try it before I give you the answers. Then match your answers with mine on the next slides. Remember: You the last digits of the deviations should come up with complimentary numbers such as: 4 & 6, 8 & 2, or 5 & 5. 40 28.6 – = 49 28.6 – = 20 28.6 – = 28 28.6 – = 38 28.6 – =

Standard deviation: step two: calculating deviations Let’s order the set vertically to make it easier: The deviation is the difference between the observation and the mean. observation – mean = deviation. Observations Mean Deviation 22 28.6 – -6.6 = 15 28.6 = -13.6 – So, the deviations for set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } are: -6.6, -13.6, 6.4, -4.6, -13.6, 11.4, 20.4, -8.6, -0.6, and 9.4. 35 28.6 = 6.4 – 24 28.6 = -4.6 – 15 28.6 -13.6 – = 40 28.6 – = 11.4 49 28.6 – = 20.4 20 28.6 – = -8.6 28 28.6 -0.6 – = 38 28.6 9.4 – =

Standard deviation: step three: squaring deviations Deviation Squared The next step is to square the deviations: deviation2 = deviation • deviation Deviation Deviation -6.6 -6.6 • = -13.6 -13.6 = • So, the deviations for set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } are: -6.6, -13.6, 6.4, -4.6, -13.6, 11.4, 20.4, -8.6, -0.6, and 9.4. 6.4 6.4 • = -4.6 -4.6 • = -13.6 • -13.6 = 11.4 11.4 • = 20.4 20.4 Try it before I give you the answers. Then match your answers with mine on the next slide. Remember: After you square the deviations, they should all end in the same number. • = -8.6 -8.6 • = -0.6 -0.6 • = 9.4 9.4 • =

Standard deviation: step three: squaring deviations Deviation Squared The next step is to square the deviations: deviation2 = deviation • deviation Deviation Deviation -6.6 -6.6 43.56 • = -13.6 -13.6 184.96 = • So, the deviations for set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } are: -6.6, -13.6, 6.4, -4.6, -13.6, 11.4, 20.4, -8.6, -0.6, and 9.4. 6.4 6.4 40.96 • = -4.6 -4.6 21.16 • = -13.6 -13.6 184.96 • = 11.4 11.4 129.96 • = 20.4 20.4 416.16 Try it before I give you the answers. Then match your answers with mine on the next slide. Remember: After you square the deviations, they should all end in the same number. • = -8.6 -8.6 73.96 • = -0.6 -0.6 3.6 • = 9.4 9.4 88.36 • =

Standard deviation: step FOur: sum the squared deviations Deviation Squared The next step is to sum the squared deviations: This simply means to add them together. 43.56 184.96 40.96 21.16 184.96 129.96 416.16 Try it before I give you the answers. Then match your answers with mine on the next slide. 73.96 .36 88.36

Standard deviation: step FOur: sum the squared deviations Deviation Squared The next step is to sum the squared deviations: This simply means to add them together. 43.56 184.96 40.96 21.16 184.96 129.96 416.16 73.96 .36 88.36 1,184.40

Standard deviation: step Five: divide the sum by one less than the observations The next step is to divide the sum by one less than the number of observations. The result is called the variance: Since there are ten observations, this means dividing the sum by nine: 1,184.40 = 294.28 n – 1 = 9, since n = 10, 10 – 1 = 9 9 Try it before I give you the answers. Then match your answers with mine on the next slide.

Standard deviation: step Five: divide the sum by one less than the observations The next step is to divide the sum by one less than the number of observations. The result is called the variance: Since there are ten observations, this means dividing the sum by nine: 1,184.40 = 294.28 n – 1 = 9, since n = 10, 10 – 1 = 9 9

Standard deviation: step six: take the square root of the variance The next step is to take the square root of the variance: 𝟐𝟗𝟒.𝟐𝟖 = Try it before I give you the answers. Then match your answers with mine on the next slide.

Standard deviation: step six: take the square root of the variance The next step is to take the square root of the variance: 𝟐𝟗𝟒.𝟐𝟖 17.2708458 = This is the standard deviation. At this point it is fine to round to the second decimal. 17.2708458 ≈ 17.27 The standard deviation of the set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } is ≈ 17.27

≈ Summary 17.27 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 To find the standard deviation of the set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } Step 1 Find the mean. Step 2 Determine the deviations. Step 3 Square the deviations. Step 4 Sum the deviations-squared. Step 5 Divide the sum of the deviations-squared by one less than the number of observations. Step 6 Take the square-root of the result. ≈ 17.27 Step 7 The result of the square root is the standard deviation. To find the standard deviation of the set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 }

challenge Try to find the standard deviation of the set = { 22, 15, 35, 24, 15, 40, 49, 20, 28, 38 } until you get it right every time.