1. Chapter 1: Exploring Data Lesson 7: Variance and Standard Deviation Mrs. Parziale

2. Vocabulary: deviation is the ______________ between each data point and the __________. variance is roughly the _____________ of the _______________ deviations. standard deviation is the ________ _______ of the variance. difference squared rootsquare average mean

3. Why use deviation and variance? Both variance and standard deviation tell us how far data in a set is spread out. It is useful if there are too few points to make an IQR (interquartile range) accurate. Variance and standard deviation are based on the mean. IQR is based on the median.

4. STEPS TO FINDING VARIANCE AND STANDARD DEVIATION: 1. Find the mean of the data set. 2. Find the difference between each value and the mean 3. Square each deviation and add them 4.Divide the sum of the squares by n-1 (n = number of elements) This equals Variance. 5. Square root the variance. This is standard deviation.

5. Example 1: Find the standard deviation of the following set of numbers (by hand): 4, 9, 11, 15, 24, 21 mean of numbers = __________________ NumberMeanDeviationDeviation 2 SUM = _______ Divide by n - 1 = 6 -1 = 5 __________ = variance = s 2 Square root (of variance), so standard deviation = __________ = s

6. Definition for finding Variance and Standard Deviation For variance = s 2, where is the mean, n is number of elements, x is the element, and i is position of element. Standard deviation = s =

7. THE CALCULATOR CAN DO THIS FOR YOU!! Steps: 1.) STAT  1: Edit…  Enter your values into L 1 2.) STAT  Calc  1: 1-Var Stats  ENTER  ENTER S x in the calculator represents “standard deviation” NOTE: You must square this number to get variance A low standard deviation indicates that the data points tend to be very close to the mean, while high standard deviation indicates that the data are “spread out” over a large range of values.

8. DO NOT CONFUSE S x WITH x IN YOUR CALCULATOR! S x is the standard deviation of a ___________ (which is what we will be using in this class) is the standard deviation of a ____________ (which is a more general case that we will not be using in this class.) NOTE: denominator of the variance of a population is “n” rather than “n – 1” as above. sample population

9. Example 2: Using your calculator, find the variance and standard deviation of the data set {4, 9, 11, 15, 24, 21}.

11. Example 3: The dot frequency distribution below shows the number of days selected students were absent in a semester. a) What is the mean of the data? b) Find the variance and standard deviation of these data. Standard deviation (S x ) = _____________ Variance (S x 2 ) = _______________ c) What can you tell from the standard deviation?

12. Example 4: A class of students is said to be homogeneous if the students in the class are very much alike on some measure. Here are four classes of students who were tested on a 20-point spelling test. Which class is the more homogeneous with respect to spelling? How can you tell? a) b) c) d)

13. Closure What is deviation? What is variance? How do you find standard deviation? If you know the standard deviation, how can you calculate variance? If you know variance, how can you calculate standard deviation? What does the standard deviation tell you about your data?