AP Stats BW 9/19 Below is a list of gas mileage ratings for selected passenger cars in miles per gallon. Choose the correct histogram of the data. 53, 43, 89, 41, 85, 86, 91, 92, 95, 94, 86, 102,114, 30, 123

Section2.4 Part 3 Chebychev’s Theorem & Grouped Data SWBAT: Identify and analyze patterns of distributions using shape, center and spread.

Distributions: Normal vs. Not or Unknown Use EMPIRICAL RULE for normal (symmetric) distributions Use CHEBYCHEV’S THEOREM for ALL distributions.

CHEBYCHEV’S THEOREM The portion of any data set lying within k standard deviations (k>1) of the mean is at least: k represents the number of standard deviations from the mean k = 2: 75% of the data lie within 2 σ of the mean k = 3: 88.9%of the data lie within 3 σ of the mean In general, Chebychev’s Theorem gives the minimum % of data that fall within the given # of standard deviations.

CHEBYCHEV’S THEOREM - Example The age distributions for Alaska and Florida are given. Which is which? What conclusions can you reach using Chebychev’s Theorem?

CHEBYCHEV’S THEOREM – Example Solution Left Graph: μ – 2σ is negative μ + 2σ = (19.5) = 70.6; thus 75% of the population is between 0 and 70.6 years old. Right Graph: μ – 2σ is negative μ + 2σ = (24.8) = 88.8; thus 75% of the population is between 0 and 88.8 years old, ie 75% of the population is under the age of Because the population is higher and they are older, we known the right graph is Florida, and the left is Alaska We would actually get more specific data using the histogram or a relative frequency histogram.

Standard Deviation for Grouped Data Earlier we found sample mean and standard deviation by creating a frequency distribution table to organize the data then using the formula for sample standard deviation:

Standard Deviation for Grouped Data, cont’d If data sets are larger and contain repeated values, you can create a relative frequency distribution to group the data: Ex: # of children in 50 households 1, 3, 1, 1, 1, 1, 2, 2, 1,0,1,1,0,0,0,1,5, 0,3,6,3,0,3,1,1,1,1,6,0,1,3,6,6,1,2,2,3,0,1,1,4,1,1,2,2,0,3,0,2,4

Let’s use a calculator instead! Enter the values of x into L1 Enter the frequencies, f, into L2 Select STAT Select CALC  1: 1-Var Stats Enter 2 nd L1, 2 nd L2

Standard Deviation for Grouped Data, cont’d When a frequency distribution has classes, you can estimate the sample mean and standard deviation using the midpoint of each class. xxx

Using midpoints Example The graph shows the results of a survey of 1000 adults. Find the mean and standard deviation.

Using midpoints Example The graph shows the results of a survey of 1000 adults. Find the mean and standard deviation. x xx $192 per year $160.3 per year

You try…. In the example, we used as the midpoint for the class of $500+. How would the sample mean and standard deviation change if you used 650 to represent the class? Find the mean and standard deviation. Sample mean is $195.5 per year, and the sample standard deviation is about $169.5 per year.

HOMEWORK: P 92: 13, 19, 29, 37, all