Module 8: Two-Way Frequency Tables Essential Questions: How can categorical data for two categories be summarized? How can you recognize possible association and trends between two categories of categorical data?

Data Collected Categorical Variable - Usually an adjective Rarely a number Examples: Gender Race Grade in School (Freshmen, Soph, Jr., Sr.) Zip Code Quantitative Variable Always a number Must be able to find the mean of the numbers Examples: Weight Height Amount of money in wallet Age

Categorical or Quantitative? 1. Survey about whether student buy lunch from the cafeteria or bring lunch from home, doesn’t eat lunch, etc. 2. Experiment where we measure how tall a plant grows. 3. Observation where we count how many people are in each car leaving school. 4. Survey about each student’s shoe size.

Two-Way Frequency Table

Complete the Tables…

Complete the Tables…

Vocab: Relative Frequency Joint Relative Frequency: tells what proportion/percentage of the total share TWO characteristics. Example: percentage of freshmen boys. Marginal Relative Frequency: tells what proportion/percentage of the total share ONE characteristic Example: percentage of boys.

1. What percent of students are boys that prefer soccer. 2 1. What percent of students are boys that prefer soccer? 2. What percent of students prefer football? 3. What is the joint relative frequency of girl and basketball? 4. What is the marginal relative frequency of soccer?

What percentage of students are girls that like oranges? What percentage of students prefer bananas? What is the joint relative frequency of boy and apple? What is the marginal relative frequency of orange?

Vocab: Conditional Frequency Conditional Frequency: describes what proportion of the population share a 2nd characteristic given that they already have one in common. Example: What proportion of students with As are girls? What proportion of girls have blue eyes? The shared/given characteristic is always the denominator.

1. What percent of boys prefer soccer. 2 1. What percent of boys prefer soccer? 2. What percent of students that prefer football are girls? 3. What proportion of girls prefer basketball? 4. Given that the student is a boy what is the probability he prefers football?

What percentage of girls like oranges? What percentage of students that prefer bananas are boys? What proportion of boys prefer oranges? Given that the student likes apples, what is the probability that he is a boy?

Vocab: Associations in Data Association: pattern/trend apparent in two-way tables. Similar in concept to correlation. Is there a association between liking aerobic exercise and liking weight lifting?

Measures of Center & Spread Graphical Representations Module 9 Measures of Center & Spread Graphical Representations Outliers & Shape Normal Distributions

Measures of Center and Spread Essential Question: How can you describe & compare data sets?

Mean Definition: Average value in data set. How to: Add up all of the values and divide by the number of numbers in the data set.

Median Definition: Middle value when the data set is arranged in order from smallest to largest How to: Put values in order. Determine middle value. If two middle values, find average of those two numbers.

Range Definition: the difference between the largest and smallest numbers in the data set.

Quartiles Definition: Quartiles are divisions representing 25% of the data. Q1 is the 1st quartile or the median of the lower half. Q3 is the 3rd quartile or the median of the upper half. How tow: Put number sin order Find median (middle number). Find median of the lower half. Find median of upper half.

Let’s Practice… Calculate the 1st, 2nd (median) and 3rd quartiles for the following data sets: 1. 15, 17, 16, 15, 18, 19, 15, 20, 18 2. 5, 8, 9, 7, 6, 9, 8 ,7, 10, 11, 4

Interquartile Range (IQR) Definition: Interquartile Range is the distance between Q1 and Q3. Subtract Q3 – Q1.

1. Find the median, IQR and range. 21, 31, 26, 24, 28, 26

2. Find the median, IQR and range.

Mrs. Daniel AP Stats- Standard Deviation Review Definition: Standard deviation is used to tell how measurements for a group are spread out from the mean. Low standard deviation = data points tend to be very close to the mean. High standard deviation value = data points are spread out

Standard Deviation

Rank the Standard Deviations:

Rank the Standard Deviations:

Rank the Standard Deviations:

Center vs. Spread Center Spread

Graphical Displays of Data Essential Question: How can we display data graphically? How can we determine the mean, median and IQR from a graphical display?

Displaying Quantitative Data Histogram Boxplot Dotplot

Number of Goals Scored Per Game by the 2004 US Women’s Soccer Team Dotplots Each data value is shown as a dot above its location on a number line. Number of Goals Scored Per Game by the 2004 US Women’s Soccer Team 3 2 7 8 4 5 1 6

How to Make a Dotplot Draw a horizontal axis (a number line) and label it with the variable name. Scale the axis from the minimum to the maximum value. Mark a dot above the location on the horizontal axis corresponding to each data value.

Let’s Practice… Create a dot plot. 12 employees at a small company make the following annual salaries (in thousands of dollars): 25, 30, 35, 35, 35, 40, 40, 40, 45, 45, 50 and 60.

Histograms Looks like a bar graph, but the bars must touch! X-axis labeled with number ranges

How to Make a Histogram Divide the range of data into classes of equal sizes. Find the count (frequency) of individuals in each class. Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals.

The histogram shows the length, in days, of Maria’s last vacations The histogram shows the length, in days, of Maria’s last vacations. Estimate the mean of the data set displayed in the histogram.

Box and Whisker Plots A box plot is a graphical display of the minimum, first quartile, median, third quartile, and maximum. The term "box plot" comes from the fact that the graph looks like a rectangle with lines extending from the top and bottom.

How to Make a Box & Whisker Plot Find the median. Find Q1 and Q3. Plot the following points: minimum, Q1, median, Q3 and maximum. Make a whisker from minimum to Q1. Make a box from Q1 to Q3. Draw a line for the median. Make a whisker from Q3 to maximum.

Create a Box and Whisker Plot. Let’s Practice: Create a Box and Whisker Plot. Quiz Scores: 25 30 26 29 22 23 24 28

Create a box plot:

Let’s Practice… Identify the median, Q1 and Q3, and the IQR. 1. 2.

Let’s Practice… Identify the median, Q1 and Q3, and the IQR. If you had to pick one career, which one would you pick and why? (Must be a statistical reason!) In tens of thousands of dollars.

Outliers & Shape Essential Question: What statistics are most affected by outliers? What shapes can data distributions have?

Outlier Outlier: a value that substantially varies from the other values. It is either much larger or much smaller.

Outliers Affect… Measure Yes/No Why? Mean Median IQR Standard Deviation Range

Let’s Practice.. Mr. Morris gave his algebra class a test, the results of which are listed below. 68, 92, 74, 75, 86, 90, 92, 81, 60, 82, 77, 80 Shania was absent on the day of the test and had to take the test late. She earned a score of 99. Which measure of the class's test results did Shania's score most change? A. IQR B. Mean C. Median D. Range

Calculate the mean, median & IQR…

Calculate the mean, median & IQR… Median: IQR: Compare:

Different Shapes of Distributions Distributions can be described as: Roughly symmetric Skewed right Skewed left

Mrs. Daniel AP Stats Shape Definitions: Symmetric: if the right and left sides of the graph are approximately mirror images of each other. Skewed to the right (right-skewed) if the right side of the graph is much longer than the left side. Skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right

Skew in Box Plots

Describe the Shape…

Describe the Shape…

Other Ways to Describe Shape: Unimodal Bimodal Multimodal

Measures of Center Type of Distribution Best Measure of Center Measures of Center = Mean and Median Type of Distribution Best Measure of Center Symmetric Skewed Right Skewed Left

Why?!?!

Which Measure of Center?

Distribution Picture Describe Best Measure of Spread? Symmetric Right Skew Left Skew

Normal Distributions Essential Question: How can you use characteristics of a normal population to make estimates and probability predictions about the population that the data represent?

Vocab: Normal Distribution Normal distribution: special type of data distribution that allows us to make predictions and estimate probabilities. Normal distributions are symmetric about the mean. Common: Heights, weights, tests scores, etc.

Normal Curve Properties

Normal Curve Properties _______ % of the data falls within 1 standard deviation of the mean. _______ % of the data falls within 2 standard deviations of the mean. _______ % of the data falls within 3 standard deviations of the mean.

How to Solve Normal Distribution Questions: Draw a curve. Label mean and standard deviation. Add and subtract standard deviation three times to label each division. Compute.

Normal Distribution Curves Pennies manufactured in the United States after 1982 have a Normally distributed mass of 2.50 g with a standard derivation of 0.02g.

Normal Distribution Curves Pennies manufactured in the United States after 1982 have a Normally distributed mass of 2.50 g with a standard derivation of 0.02g. What proportion of pennies have a mass greater than 2.50g? What proportion of pennies have a mass between 2.48g and 2.52g? What proportion of pennies have a mass less than 2.48? What proportion of pennies have a mass between 2.46 and 2.50g?

Normal Distribution Curves 15 year-old girls have heights that are Normally distributed with a mean of 64 inches and a standard deviation of 1.5 inches.

Normal Distribution Curves 15 year-old girls have heights that are Normally distributed with a mean of 64 inches and a standard deviation of 1.5 inches. What proportion of girls are taller than 65.5 inches? What proportion of girls are shorter than 64 inches? What proportion of girls have a height between 61 and 65.5 inches?