Bell Work 1. 2. 3.

Bell Work Identify the least and greatest value in each data set. 1. 34, 62, 45, 35, 75, 23, 35, 65, 23 2. 1.6, 3.4, 2.6, 4.8, 1.3, 3.5, 4.0 Order the data from least to greatest and find mean median and mode. 3. 2.4, 5.1, 3.7, 2.1, 3.6, 4.0, 2.9 4. 5, 5, 6, 8, 7, 4, 6, 5, 9, 3, 6, 6, 9 23, 75 1.3, 4.8 2.1, 2.4, 2.9, 3.6, 3.7, 4.0, 5.1 3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 9, 9

Data Analysis

Objectives The student will be able to: find the variance of a data set. find the standard deviation of a data set. SOL: A.9 2009

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

Variance 1. Find the mean of the data. Hint – mean is the average so add up the values and divide by the number of items. 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 the number of items.

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.

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 is often denoted by the lowercase Greek letter sigma, .

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.

Standard Deviation Find the variance. 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 the number of items. Take the square root of the variance.

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

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

Find the variance and 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 to find the variance: 1056/5 = 211.2

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

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 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: 92,92,92,52,52. Find the standard deviation for this class. Answer Now

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 : 1920/5 = 384 variance 6) Find the square root of the variance: Thus the standard deviation of the second set of test scores is 19.6.

Analyzing the data: Consider both sets of scores. Both classes have the same mean, 76. However, each class does not have the same scores. Thus we use the standard deviation to show the variation in the scores. With a standard variation of 14.53 for the first class and 19.6 for the second class, what does this tell us? Answer Now

Analyzing the data: Class A: 92,88,80,68,52 Class B: 92,92,92,52,52 With a standard variation of 14.53 for the first class and 19.6 for the second class, the scores from the second class would be more spread out than the scores in the first class.

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

Bell Work 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.

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.

Frequency Tables

Frequency Tables Statistics deals with collecting, organizing, and interpreting data. Data are pieces of information, which are often numerical. Large amounts of data can be organized in a frequency table. A frequency table shows the number of pieces of data that fall within given intervals.

Frequency Tables Words to Know Price $ Tally Frequency 1-25 4 26-50 11 51-75 7 76-100 13 Shows the number of pieces of data that fall within given intervals. Scale- includes the least value and greatest value Interval- Separates the scale into equal parts

Make a frequency table Step 1- Choose an appropriate interval and scale. Should include the least and the greatest value. Step 2- Draw a table with 3 columns and label the columns. Step 3- Complete the table.

Example Scores Tally Frequency Winning Score 20 34 23 34 31 35 27 49 20 34 23 34 31 35 27 49 30 52 37 20 55 20 42 39 46 38 38 27 20 - 29 6 9 30 - 39 40 - 49 3 50 - 59 2

Example Music Tally Frequency Favorite Music P R F P F F R F P P F P F C P C J R R F J R P P F R= rock J= jazz C= country P= rap F= top 40 Rock 5 2 Jazz Country 2 Top 40 8 Rap 8

Example – Your Turn Rentals Tally Frequency Movie Rentals 10 9 2 17 12 10 9 2 17 12 9 11 10 4 14 3 15 0 6 5 9 20 10 11 8 3 1 13 4 5

Bell work   Which of the following data sets has the smallest standard deviation? a. 51,52,52,50,50,53,49 b. 55,58,52,50,22,35,42 c. 37,41,33,31,12,24,43 d. 28,34,24,21,8,36,13,

Bell Work 1. 2. 3. 5. 4.