Dotplots Horizontal axis with either quanitative scale or categories Each dot represent one data point If more than one data point has the same value, “stack” the dots CASA, AP Stats 05-06, Section 1.1.1

Stemplots Each data point is represented by a single number (leaf) associated with a more significant number (stem) Rounding can help the shape of the distribution become more noticeable. CASA, AP Stats 05-06, Section 1.1.1

Histograms Each row of a stemplot represents a number of data points found in a range. What kind of ranges might a stemplot have? Histograms have the same general idea, but histograms are more flexible in terms of the ranges. Ranges have equal width CASA, AP Stats 05-06, Section 1.1.1

Histograms Histograms may be less useful than stemplots because you can’t see the actual data. CASA, AP Stats 05-06, Section 1.1.1

Histograms Ranges for histograms generally include the lower end of the range, but does not include the upper end of the range CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms The TI-83 can be used to make histograms, and will allow you to change the location and widths of the ranges. CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms Start by entering data into a list Example: Enter the presidential data from page 19 into any list; in this case, we will use L1 CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms Choose 2nd:Stat Plot to choose a histogram plot Caution: Watch out for other plots that might be “turned on” or equations that might be graphed CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms Turn the plot “on”, Choose the histogram plot. X-list should point to the location of the data. CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms Under the “Zoom” menu, choose option 9: ZoomStat CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms The result is a histogram where the calculator has decided the width and location of the ranges You can use the Trace key to get information about the ranges and the frequencies CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms You can change the size and location of the ranges by using the Window button Press the Graph button to see the results CASA, AP Stats 05-06, Section 1.1.1

Using the TI-83 to make histograms Voila! Of course, you can still change the ranges if you don’t like the results. CASA, AP Stats 05-06, Section 1.1.1

Using Percents It is sometimes difficult to compare the “straight” numbers. For example: Ty Cobb in his 24 year baseball career had 4,189 hits. Pete Rose also played for 24 years but he collected 4,256 hits. Was Pete Rose a better batter than Ty Cobb? CASA, AP Stats 05-06, Section 1.1.1

Using Percents Ty Cobb had a career batting average of .366. (He got a hit 36.6% of the time) Pete Rose had a career batting average of .303. Comparing the percentage, we get a clearer picture. CASA, AP Stats 05-06, Section 1.1.1

Relative Frequency When constructing a histogram we can use the “relative frequency” (given in percent) instead of “count” or “frequency” Using relative frequency allows us to do better comparisons. Histograms using relative frequency have the same shape as those using count. CASA, AP Stats 05-06, Section 1.1.1

Finding Relative Frequency For each count in a class, divide by the total number of data points in the data set. Convert to a percentage. CASA, AP Stats 05-06, Section 1.1.1

Finding Relative Frequency Class Frequency 40-44 2 45-49 6 50-54 13 55-59 12 60-64 7 65-69 3 Total 43 CASA, AP Stats 05-06, Section 1.1.1

Finding Relative Frequency Class Frequency Relative Frequency 40-44 2 2/43=4.7% 45-49 6 6/43=14.0% 50-54 13 13/43=30.2% 55-59 12 12/43=27.9% 60-64 7 7/43=16.3% 65-69 3 3/43=7.0% Total 43 CASA, AP Stats 05-06, Section 1.1.1

Histograms CASA, AP Stats 05-06, Section 1.1.1

Finding Cumulative Frequency Class Frequency Relative Frequency Cumulative Frequency 40-44 2 2/43=4.7% 45-49 6 6/43=14.0% 8 50-54 13 13/43=30.2% 21 55-59 12 12/43=27.9% 33 60-64 7 7/43=16.3% 40 65-69 3 3/43=7.0% 43 Total CASA, AP Stats 05-06, Section 1.1.1

Finding Relative Cumulative Frequency Class Frequency Relative Frequency Cumulative Frequency Relative Cumulative Frequency 40-44 2 2/43=4.7% 45-49 6 6/43=14.0% 8 8/43=18.6% 50-54 13 13/43=30.2% 21 21/43=48.8% 55-59 12 12/43=27.9% 33 33/43=76.7% 60-64 7 7/43=16.3% 40 40/43=93.0% 65-69 3 3/43=7.0% 43 43/43=100% Total CASA, AP Stats 05-06, Section 1.1.1

Percentiles “The p-th percentile of a distribution is the value such that p percent of the observations fall at or below it.” If you scored in the 80th percentile on the SAT, then 80% of all test takers are at or below your score. CASA, AP Stats 05-06, Section 1.1.1

Relative Cumulative Frequency Percentiles Class Relative Cumulative Frequency 40-44 2/43=4.7% 45-49 8/43=18.6% 50-54 21/43=48.8% 55-59 33/43=76.7% 60-64 40/43=93.0% 65-69 43/43=100% Total It is easy to see the percentiles at the breaks. “A 64 year old would be at the 93rd percentile.” What do you do for a 57 year old? CASA, AP Stats 05-06, Section 1.1.1

Ogives or “Relative Cumulative Frequency Graph” Class Relative Cumulative Frequency 40-44 2/43=4.7% 45-49 8/43=18.6% 50-54 21/43=48.8% 55-59 33/43=76.7% 60-64 40/43=93.0% 65-69 43/43=100% Total CASA, AP Stats 05-06, Section 1.1.1

AP Statistics September 3, 2014 Mr. Calise Section 1.2 Part 1 AP Statistics September 3, 2014 Mr. Calise

Describing Distributions with Numbers: Center/Mean The Mean The Average The Arithmetical Mean The mean is not a resistance measure of center AP Statistics, Section 1.2, Part 1

Describing Distributions with Numbers: Center/Median When numbers are ordered from low to high, the median is the middle number (if n is odd) or the average of the two middle numbers (if n is even) Resistant measure of center AP Statistics, Section 1.2, Part 1

Describing Distributions with Numbers: Spread/IQR The first quartile (Q1) is the median of the first half of the distribution The third quartile (Q3) is the median of the second half of the distribution The interquartile range (IQR) is the distance between the first and third quartiles. (IQR = Q3 – Q1) AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Finding M, Q1, Q3, IQR Barry Bonds’ home run counts per season (average=35.4): 16, 19, 24, 25, 25, 33, 33, 34, 34, 37, 37, 40, 42, 46, 49, 73 Hank Aaron’s home run counts per season (average=34.9): 13, 20, 24, 26, 27, 29, 30, 32, 34, 34, 38, 39, 39, 40, 40, 44, 44, 44, 44, 45, 47 AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Finding M, Q1, Q3, IQR Barry Bonds: Q1=25, M=34, Q3=41, IQR=16 Hank Aaron: Q1=28, M=38, Q3=44, IQR=16 What does this say about comparing Bonds and Aaron? AP Statistics, Section 1.2, Part 1

Describing Distributions with Numbers: Outliers If a data point falls more than 1.5 * IQR above the Q3 or falls less than 1.5 * IQR below the Q1. Barry Bonds: M=34, Q1=25, Q3=41, IQR=16 41 + 1.5 * 16 = 65 (anything above 65) 25 - 1.5 * 16 = 1 (anything below 1) AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Side by Side Boxplots AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Modified Boxplot AP Statistics, Section 1.2, Part 1

Describing with Numbers: Spread/Standard Deviation s2 is called the variance, and its in units2 s is called the standard deviation, and its in units. AP Statistics, Section 1.2, Part 1

Describing with Numbers: Spread/Standard Deviation s measures spread about the mean and should be used only when the mean is chosen as the measure of center s equals 0 only when there is no spread. This happens only when all observations have the same value. s is not resistant. Outliers effect it adversely. AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Calculating s by hand Observation (x-bar) Deviation (x-sub-i – x-bar) Squared Deviation 1792 1666 1362 1614 1460 1867 1439 Sum = 11200 Mean = 1600 AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Calculating s by hand Observation (x-bar) Deviation (x-sub-i – x-bar) Squared Deviation 1792 1792-1600 = 192 1666 1666-1600 =66 1362 1362-1600 =-238 1614 1614-1600 =14 1460 1460-1600 =-140 1867 1867-1600 =267 1439 1439-1600 =-161 Sum = 11200 Mean = 1600 Sum = 0 AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Calculating s by hand Observation (x-bar) Deviation (x-sub-i – x-bar) Squared Deviation 1792 1792-1600 = 192 192 * 192 = 36864 1666 1666-1600 =66 4356 1362 1362-1600 =-238 56644 1614 1614-1600 =14 196 1460 1460-1600 =-140 36864 1867 1867-1600 =267 71289 1439 1439-1600 =-161 25921 Sum = 11200 Mean = 1600 Sum = 0 Sum = 214870 AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Calculating s by hand AP Statistics, Section 1.2, Part 1

Calculating s using TI-83 AP Statistics, Section 1.2, Part 1

Calculating s using TI-83 AP Statistics, Section 1.2, Part 1

Calculating s using TI-83 AP Statistics, Section 1.2, Part 1

Linear Transformations Adding “a” does not change the shape or spread Adding “a” does change the center by “a” AP Statistics, Section 1.2, Part 1

Linear Transformations Multiplying by “b” does change the shape or spread by “b” Multiplying by “b” does change the center by “b” AP Statistics, Section 1.2, Part 1

Comparing Distributions We have already seen the usefulness of comparing two distribution in the answering difficult questions like: “Who was a better home run hitter, Barry Bonds or Hank Aaron?” We used side-by-side boxplots to answer that question. AP Statistics, Section 1.2, Part 1

Back-to-Back Stemplots We can use stemplots as a way of doing comparisons between two distributions. For example, which baseball league is more competitive, the American League or the National League? AP Statistics, Section 1.2, Part 1

Back-To-Back Stemplots AP Statistics, Section 1.2, Part 1

Back-To-Back Stemplots AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Reminders If there are outliers, don’t use mean or standard deviation to compare distributions. You can’t compare a standard deviation of one distribution to the IQR of a different distribution. You can’t compare a mean of one distribution to the median of a different distribution. AP Statistics, Section 1.2, Part 1

AP Statistics, Section 1.2, Part 1 Homework Assignment Exercises 1.31 (a) & (b) 1.33, 1.34, 1.35 1.36 (a) AP Statistics, Section 1.2, Part 1