Standard Deviation Standard Deviation summarizes the amount each value deviates from the mean. SD tells us how spread out the data items are in our data set.

If data is close together, standard will be small. If the data is spread out, standard deviation will be large. Standard Deviation = sigma, σ

Standard Deviation Find the standard deviation. a) Find the mean of the data. b) Subtract the mean from each value. c) Square each deviation of the mean. d) Find the sum of the squares. e) Divide the sum by the number of items. (average) f. Take the square root.

Standard Deviation Formula This is what we just did! = find the sum x = your data point = the mean n = the number of items.

Standard deviation shows how spread out our data is SD shows how consistent our data is SD shows how much our data deviates from the mean

Find the standard deviation The math test scores of five students are: 92,88,80,68 and 52. 1) Find the mean: (92+88+80+68+52)/5 = 76. 2) Find each deviation from the mean: 92-76=16 88-76=12 80-76=4 68-76= -8 52-76= -24

Find the standard deviation The math test scores of five students are: 92,88,80,68 and 52. 3) Square the deviations from the mean:

standard deviation The math test scores of five students are: 92,88,80,68 and 52. 4) Find the sum of the squares of the deviation from the mean: 256+144+16+64+576= 1056 5) Divide by the number of data items : 1056/5 = 211.2

Find the standard deviation The math test scores of five students are: 92,88,80,68 and 52. 6) Find the square root: standard deviation of the test scores is 14.53.

Standard Deviation A different math class took the same test with these five test scores: 76,88,82,68,66. Find the standard deviation for this class.

Hint: Find the mean of the data. Subtract the mean from each value – called the deviation from the mean. Square each deviation of the mean. Find the sum of the squares. Divide the total by the number of items – result is the variance. Take the square root of the variance – result is the standard deviation.

Solve: A different math class took the same test with these five test scores: 76,88,82,68,66. Find the standard deviation for this class. Answer Now

The math test scores of five students are: 76,88,82,68,66 1) Find the mean: (76+88+82+68+66)/5 = 76 2) Find the deviation from the mean: 76-76=0 88-76=12 82-76=6 68-76= -8 66-76= -10 3) Square the deviation from the mean: 0 144 36 64 100 4) Find the sum of the squares: 0+144+36+64+100= 344

The math test scores of five students are: 76,88,82,68,66. 5) Divide the sum of the squares by the number of items : 344/5 = 68.8 6) Find the square root of the variance: square root of 68.8 =8.3 Thus the standard deviation of the second set of test scores is 8.3.

But, each class does not have the same scores. Analyzing the data: Both classes a mean of, 76. But, each class does not have the same scores. Thus we use the SD to see how the scores deviate from the mean. 1st class SD =14.53 2nd class SD =8.3 What does this tell us? Answer Now

Analyzing the data: Class A: 92,88,80,68,52 Class B: 76,88,82,68,66 1st class 14.53 2nd class 8.3 the scores from the second class would be more consistent than the scores in the first class OR less spread out.

b) Standard deviation will be greater than 19.6. Analyzing the data: Class A: 92,88,80,68,52 Class B: 92,92,92,52,52 Class C: 77,76,76,76,75 Estimate the standard deviation for Class C. a) Standard deviation will be less than 14.53. b) Standard deviation will be greater than 19.6. c) Standard deviation will be between 14.53 and 19.6. d) Can not make an estimate of the standard deviation. Answer Now

The bell curve, which represents a normal distribution of data shows what standard deviation represents. One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99 percent of the data.

Answer: A Analyzing the data: Class C: 77,76,76,76,75 Class A: 92,88,80,68,52 Class B: 92,92,92,52,52 Class C: 77,76,76,76,75 Estimate the standard deviation for Class C. a) Standard deviation will be less than 14.53. b) Standard deviation will be greater than 19.6. c) Standard deviation will be between 14.53 and 19.6 d) Can not make an estimate if the standard deviation. Answer: A The scores in class C have the same mean of 76 as the other two classes. However, the scores in Class C are all much closer to the mean than the other classes so the standard deviation will be smaller than for the other classes.

SD tells the researcher how spread out the responses are -- are they concentrated around the mean, or scattered far & wide?Did all of your respondents rate your product in the middle of your scale, or did some love it and some hate it?

Let's say you've asked respondents to rate your product on a series of attributes on a 5-point scale. Mean for "good value for the money" was 3.2 with a SD of 0.4 Mean for "product reliability" was 3.4 with a SD of 2.1. Looking at the means only reliability was rated higher than value. But the higher SD for reliability could indicate that responses were more spread out (good and bad ratings).

Another way of looking at Standard Deviation is by plotting the distribution as a histogram of responses. A distribution with a low SD would display as a tall narrow shape, while a large SD would be indicated by a wider shape.

Summary: As we have seen, standard deviation measures the dispersion of data. The greater the value of the standard deviation, the further the data tend to be dispersed from the mean.