Objectives The student will be able to:

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Presentation transcript:

Objectives The student will be able to: find the variance of a data set. find the standard deviation of a data set.

Definition of Variance Variance is the average squared deviation from the mean of a set of data. It is used to find the standard deviation.

Finding The Variance 1. Find the mean of the data. Subtract the mean from each value – the result is called the deviation from the mean. 3. Square each deviation of the mean. 4. Find the sum of the squares. 5. Divide the total by (n-1).

Variance Formula The variance formula includes the Sigma Notation which represents the sum of all the items to the right of Sigma. Mean is represented by and n is the number of items.

Definition of Standard Deviation Standard Deviation shows the variation in data. If the data is close together, the standard deviation will be small. If the data is spread out, the standard deviation will be large.

Standard Deviation Find the variance first! 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 total by (n – 1) Take the square root of the variance = Sx

Standard Deviation Formula The standard deviation formula can be represented using Sigma Notation: Notice the standard deviation formula is the square root of the variance.

Find the variance and 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 the deviation from the mean: 92-76=16 88-76=12 80-76=4 68-76= -8 52-76= -24

Example 3) Square the deviation from the mean:

Example 4) Find the sum of the squares of the deviation from the mean: 256+144+16+64+576= 1056 5) Divide by (n-1) to find the variance: 1056/4 = 264

Standard Deviation . 6) Find the square root of the variance: Thus the standard deviation of the test scores is 16.25

Standard Deviation A different math class took the same test with these five test scores: 92,92,92,52,52. 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 (n-1) to get the variance. Take the square root of the variance to find the standard deviation.

The math test scores of five students are: 92,92,92,52 and 52. 1) Find the mean: (92+92+92+52+52)/5 = 76 2) Find the deviation from the mean: 92-76=16 92-76=16 92-76=16 52-76= -24 52-76= -24 3) Square the deviation from the mean: 4) Find the sum of the squares: 256+256+256+576+576= 1920

The math test scores of five students are: 92,92,92,52 and 52. 5) Divide the sum of the squares by the number of items : 1920/4 = 480 variance 6) Find the square root of the variance: Thus the standard deviation of the second set of test scores is 21.91.

Standard Deviation in the Calculator! Data values in first list: 92,92,92,52,52

Standard Deviation in the Calculator!

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.