Chapter 3: Section 2 Measures of Variance

Paint Comparison: How many months will they last??? Brand ABrand B Average for Brand A: Average for Brand B: 210/6 = 35 months Find the mean of each Brand.

Lets Check the Range: Range for Brand A: Range for Brand B: 60 – 10 = 50 Months 45 – 25 = 20 Months

Finding Variance and Standard Deviation of a Population Variance Formula for a Population Standard Deviation Formula for a Population X = individual value μ = population mean N = population size

μ= 35 N = 6 Finding Variance and Standard Deviation of a Population for Brand A 1750Total (X – μ)^2X - μX

Finding Variance and Standard Deviation of a Population for Brand A Variance is 291.7Standard Deviation is 17.1

μ= 35 N = 6 Finding Variance and Standard Deviation of a Population for Brand B 250Total (X – μ)^2X - μX

Finding Variance and Standard Deviation of a Population for Brand B Variance is 41.7Standard Deviation is 6.5

Which Brand is right for you? Brand A Variance: Standard Deviation: 17.1 Brand B Variance: 41.7 Standard Deviation: 6.5

Page 137 # 6

Finding Variance and Standard Deviation of a Sample Variance Formula for a Sample Standard Deviation Formula for a Sample X = individual value = sample mean n = sample size

Page 129 Example , 11.9,12.0, 12.8, 13.4, 14.3

Finding Variance and Standard Deviation of a Sample Variance is 1.28 Standard Deviation is 1.13

= 12.6 n = 6 Finding Variance and Standard Deviation of a Sample using Chart 6.38Total (X – )^2X -X

Finding Variance and Standard Deviation of a Sample using Chart Variance is 1.28 Standard Deviation is 1.13

Page 138 # 10

Variance and Standard Deviation from a Frequency Table 13,310490Total: 20 2, , , , Fr *(MP^2)Freq * MPMidpointFrequencyClass

Formula Variance is 68.7 Standard Deviation is 8.3

Page 139 # 27

91,3822,733Total: 80 3, , , ,8481, , , Fr *(MP^2)Freq * MPMidpointFrequencyClass Page 139 # 27

Formula Variance is 68.7 Standard Deviation is 8.3

Standard Deviation 1 Standard Deviation above and below the Mean 2 Standard Deviation above and below the Mean

Chebyshev’s Theorem The proportion of values from the a data set that fall within k standard deviations of the mean will be at least 1-1/k², where k is a number greater then 1 (k is not necessarily an integer). Basically, it states that: At least 75% of the data will fall within 2 Standard Deviations from the mean At least 88.89% if the data will fall within 3 Standard Deviations of the mean.

Proof of 75% and 88.89%: 1-1/k² can be written as: 1-1/2² = 1-1/4 = ¾ = 75% 1-1/3² = 1-1/9 = 8/9 = 88.89%

Example: The mean price of a house in a certain neighborhood is $50,000 and the standard deviation is $10,000. Find the price range for which at least 75% of the houses will sell.

Page 135 Example 3-27 Since 75% of the data fall with in two standard deviations above and below the mean: Use the following formulas: Above = mean + 2 standard Deviations times the standard deviation Below = mean - 2 standard Deviations times the standard deviation

Page 135 Example 3-27 Therefore: 50, (10,000) = $70,000 50,000 – 2(10,000) = $30,000 Hence, at least 75% of the houses will sell between $30,000 and $70,000.

Chebyshev and Percents: A survey of local companies found that the mean amount of travel allowance for executives was $0.25 per mile. The standard deviation was $0.02. Using Chebyshev’s Theorem, find the minimum percentage of data values that will fall between $0.20 and $0.30.

Using Chebyshev Step 1: Subtract the mean from the larger value: $ $0.25 = $0.05 Step 2: Divide the difference by the standard deviation to get k. k = 0.05/0.02 = 2.5 Step 3: Use Chebyshev Theorem to find percent: 1-1/k² = 1-1/2.5² = 0.84

Conclusion: 84% of the data will fall between $0.20 and $0.30.

Class/Homework Page: #’s and 45

Finding Q1, Q2, and Q3 89, 47, 164, 296, 30, 215, 138, 78, 48, 39 Step 1: Arrange Data in ascending order 30, 39, 47, 48, 78, 89, 138, 164, 215, 296 Step 2: Find Q2 a.k.a. the MEDIAN 30, 39, 47, 48, 78, 89, 138, 164, 215, 296 (78+89)/2 = 83.5

Continued Step 3: Find Q1 30, 39, 47, 48, 78, Q1 = 47 Step 4: Find Q3 Q3 = , 138, 164, 215, , 39, 47, 48, 78,89, 138, 164, 215, 296

Making a Box Plot (5 pt summary) Step 1: Find the following Values: Min = Q1 = Med = Q3 = Max =

Making a Box Plot (5 pt summary) Step 2: Draw a number line Step 3: Draw 5 pts Step 4: Draw box around Q1 and Q3 Step 5: Draw line on Median Step 6: Connect max and min to box

Practice: Page 175 Problems: 21 and 22 Four Box plots

Using the z score A student scored 38 on a Calculus test that had a mean of 40 and a standard deviation of 5; she scored 94 on a history test with a mean of 100 and a standard deviation of 10. Compare her relative position on the two tests.

Chapter 3 Section 3: From a to z score The z score (standard score) is obtained by subtracting the mean from the value and dividing by result by the standard deviation. For SampleFor Population Words Formula

Using the Formula s= 10Mean = 100X = 94Test B s= 5Mean = 40X = 38Test A Test B

Try: Page 153 #’s 9-15

Finding Your Percentile A teacher gives a 20-pt test to 10 students. The scores are: 18, 15, 12, 6, 8, 2, 3, 5, 20, 10 Find the percentile rank of a score of 12.

Steps for Percentiles Step 1: Arrange the Data in accending order 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 Step 2: Use Percentile Formula Percentile = (Number of values below X) Number of Values below X: Total Number of values *

Percentile Continued….. Therefore a score of 12 was better than 65% of the class.

Find the score of a given Percentile Step 1: Arrange Data in increasing order Step 2: Use Formula: Where: n = total number of values p = percentile Step 3A: If c is not a whole #, round up to the next number and count over from smallest #. Step 3B: If c is a whole #, count over from the smallest #. Then add that number to the next number and divide by two

Example: What score represents the 60 th percentile? 2, 3, 5, 6, 8, 10, 12, 15, 18, 20 Whole # Since 6 is a whole number, count over 6 terms. Then add that # to the next # and divide by 2. The score of 11 represents the 60 th percentile.

Try: Page 154 – 155 # Even # odd