Standard Deviation, Z- Scores, Variance ALGEBRA 1B LESSON 42 INSTRUCTIONAL MATERIAL 2

Standard Deviation  Standard Deviation is a number that tells us how a value in the data set differs from the mean. It also tells us how spread out the numbers are. The bigger the standard deviation, the more widespread the data is.  The Greek letter sigma, , will represent Standard Deviation.

Standard Deviation  To find Standard Deviation, there are 2 ways:  One way is the “long” way:  Calculate the mean  Subtract the mean from each data value  Square the differences (results from 2 nd step)  Find the average of the squared differences (This is called “Variance”)  Take the square root of the variance

Example

Cont.  Subtract the mean from each data value:  75 – 83.67=  77 – =  86 – = 2.33  89 – = 5.33  93 – = 9.33  82 – = 1.67

Cont.  Square the difference and find the average: (-8.67) 2 = 75 (-6.67) 2 = 44 (2.33) 2 = 5 (5.33) 2 = 28 (9.33) 2 = 87 (1.67) 2 = 2

 The 2 nd way to find Standard Deviation (the shortcut):  Type the data values in your calculator  Stat, Edit, L 1  Stat, Calc, 1, Calculate – look for the sigma, .  Standard Deviation = 6.4 (Between the calculator and paper method, the standard deviation may be slightly different. That is due to rounding. The more places you round to, the more accurate your answer is.)

Example 2  Find the Standard Deviation for the following average temperatures: 63, 61, 66, 67, 63, 65, 68  Standard Deviation (  = 2.3)

Z-Score

Example 3

Cont.  With a z-score of -3, that means the student scored 3 standard deviations below the mean.

Example 4

Example 5

Cont.

Summary  Standard Deviation, Variance, and Z-Score are 3 different ways to analyze data.  The Standard Deviation and Variance describe how spread out data values are from each other, as well as from the mean. Given the variance, take the square root to find the standard deviation.

Summary Cont.  The Z-Score measures how far a data value is away from the mean in terms of Standard Deviation.  The Standard Deviation is what you add/subtract to the mean while the z- score is how many times you add/subtract to the mean.