Statistics. The Process:

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1 AP Statistics Chapter 1 Notes ( ) Introduction: Data Analysis: Making Sense of Data Statistics Descriptive statistics is the term given to the analysis of data that helps describe, show or summarize data in a meaningful way such that, for example, patterns might emerge from the data. Descriptive statistics do not, however, allow us to make conclusions beyond the data we have analyzed or reach conclusions regarding any hypotheses we might have made Inferential statistics are techniques that allow us to use these samples to make generalizations about the populations from which the samples were drawn The Process: 1

2 Any set of data contains information about some group of. The characteristics we measure on each individual are called. Ex.: In a study about car battery life, the individuals are and the variable is. Types of Variables Places an individual into one of several groups or categories Takes numerical values for which makes sense to take an average Examples: Examples: Every possible value can be listed There is an infinite number of values the variable can take on. ** Not every variable that takes number values is quantitative. Ex. It is common to transform a quantitative variable into a categorical variable. Ex.: 2

3 Alt. Example: Here is information about 7 randomly selected U.S. residents from the 2000 census. State Number of family members Age Gender Marital Status Total Income Kentucky 2 61 F M 21, Florida 6 27 F S 21, Wisconsin 2 27 M M 30,000 5 California 4 33 F M 526, Michigan 3 49 F M 15, Virginia 3 26 F M 25, Pennsylvania 4 44 M M 43, a) Who are the individuals in this data set? Travel time to work b) What variables are measured? Identify each as categorical or quantitative. In what units were the quantitative variables measured? c) Describe the individual in the first row. d) Describe the individual in the last row. Quantitative variables may take up values that are very close together or values that are quite spread out. We call the pattern of variation of a variable its. The of a variable tells us what values the variable takes and how often it takes these values. CYU (Pg.5) 3

4 AP Statistics Chapter 1 Notes 1.1 Analyzing Categorical Data The values of a categorical variable are. The distribution of a categorical variable lists the categories and gives either the count or the percent of individuals who fall in each category. Here is the distribution of bachelor s degrees awarded in 2007, Major Number of Majors Percent of Majors The individuals are: Business 327, Social sciences/history 164, The variable is: Education 105, Health professions 101, Psychology 90, The middle column displays: Biological and biomedical sciences Communication and related programs 75, , Engineering 67, The last column displays: English language and literature 55, Other 462, Total 1,524, **Percents should add to 100%. Sometimes, when each percent is rounded and then all percents are added, the total does not equal 100%. (Should be close to 100%). This is a and does not necessarily mean that we made a mistake in our calculations. Displaying categorical data graphically: Bar Graphs: Used to display the distribution of a variable. Easy way to compare categories. Show the for each identified category. (represented by the height of the bars.) Each category s information is depicted in a. Individual bars are separated by a. ** As with any graph presented to you to analyze, always read the first. Deceptive Graphs: Pictographs and inconsistent scales. (See pg.11 Ex. Who Buys imacs?) 4

5 Two-Way Tables and Marginal Distributions: Two-Way Tables are used when a data set involves categorical variables. They contain much more information than the two marginal distributions alone. Alt. Example: Super Powers Female Male Total Invisibility Super Strength Telepathy Fly Freeze Time Total A sample of 200 children from the U K ages 9-17 was randomly selected. The gender of each student was recorded along with which super power they would most like to have: invisibility, super strength, telepathy, ability to fly, or ability to freeze time. This is a two-way table because it describes two categorical variables, and It is OK to switch the location of the variables. a) Use the data in the two-way table to calculate the marginal distribution (in percents) of superpower preferences. (b) Make a graph to display the marginal distribution. Describe what you see. The distributions of opinion alone and gender alone are called because they appear at the right and bottom margins of the two-way table. Relationships between Categorical Variables: Conditional Distributions. 5

6 Marginal distributions tell us nothing about the relationship between two variables. To describe a relationship between the two variables, we must do some calculations. Calculate the conditional distribution of responses for the males and females. Percent of Females Percent of Males F M T Invisibility Super Strength Telepathy Invisibility Super Strength Telepathy Fly Freeze Time Fly Total Freeze Time Total A conditional distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. There is a separate conditional distribution for each value of the other variable. (i.e., Finding percentages for all five entries in the Female column gives the conditional distribution of opinion among girls. Construct a segmented bar graph of the conditional distribution of gender for each super power. I F M T SS T F FT Total Concluding: Based on the survey data, can we conclude that boys and girls differ in their preference of superpower? Give appropriate evidence to support your answer. We say that there is an between two variables if specific values of one variable tend to occur in common with specific values of the other. **Even strong association between two categorical variables can be influenced by other variables lurking in the background. 6

7 AP Statistics Chapter 1 Notes 1.2 Displaying Quantitative Displaying data graphically: Dotplots, stemplots, and Histograms Describing the distribution of quantitative variables: S O C S A distribution is: Roughly (approximately) symmetric if the right and left sides of the graph are approximately mirror images of each other. Skewed to the right if the right side of the graph is much longer than the left side. Skewed to the left if the left side of the graph is much longer than the right side. Unimodal: Single peak (ONE MODE) Bimodal: two peaks (TWO MODES) Multimodal: More than two clear peaks. (MORE THAN TWO MODES) Not all distributions have a simple shape, especially when there are few observations. This is a departure from the general pattern. It refers to an individual value that falls outside the overall pattern (much larger/smaller than the rest of the observations) Use mean or median to describe center. This tells you how much variability there is in the data. Use range or interquartile range. **AND ALWAYS USE CONTEXT DOTPLOTS: Simple graph used to display quantitative data. Each data value is shown as a dot plot above its location on a number line. Make it easy to describe the distribution of quantitative variables. DON T FORGET TO: Label the number line, have consistent, clear intervals, and place the points correctly. Alt. Example: Oops! We lost the ball Here are the number of turnovers for the Oakland Raiders football team during their 16 regular season NFL games: 3, 0, 3, 3, 3, 2, 4, 1, 2, 3, 1, 0, 1, 2, 3, 2. Make a dotplot to display the data. 7

8 STEMPLOTS: (also stem-and-leaf plots) Another simple display of quantitative data. Use it for data sets. Gives a quick picture of the shape of a distribution while including the actual numerical values in the graph. Each observation is separated into a stem and a one-digit leaf Do not work well with large data sets, where each stem must hold a large number of leaves. Use as a minimum number of stems. Too few or too many make it difficult to see the distribution s shape. You can split stems, but be sure that each stem is assigned an equal number of possible leaf digits. You may or the data if the data have too many digits. Use a back-to-back stemplot with common stems when you want to compare sets of data. Example: How Many Shoes? How many pairs of shoes does a typical teenager have? To find out, a group of AP Statistics students conducted a survey. They selected a random sample of 20 female students from their school. Then they recorded the number of pairs of shoes that each respondent reported having. Here are the data: 50, 26, 26, 31, 57, 19, 24, 22, 23, 38, 13, 50, 13, 34, 23, 30, 49, 13, 15, 51 A random sample of 20 male students at the same school: 14, 7, 6, 5, 12, 38, 8, 7, 10, 10, 10, 11, 4, 5, 22, 7, 5, 10, 35, 7 Make a stemplot to display the data: Steps: 1) Separate each observation into a STEM, consisting of all but the final digit, and a LEAF, the final digit. Write the stem in a vertical column with the smallest at the top, and draw a vertical line at the right of this column. **DO NOT SKIP ANY STEMS, even if there is no data value for a particular stem. 2) Write each leaf in the row to the right of its stem. Arrange the leaves in increasing order out from the stem. 3) Provide a key that explains in context what the stems and leaves represents. 4) Check your work! Describe the distribution if asked. 8

9 HISTOGRAMS: Most common graph of the distribution of one quantitative variable. Use relative frequency ( on the vertical axis) histograms when comparing data sets of different sizes. Make sure to have a minimum of classes. Ex. Creating a histogram. David handles his own portfolio and has done so for years. Below is the number of years David keeps a stock before selling it: 3, 3, 8, 8, 8, 9, 9, 10, 11, 11, 11, 11, 11, 12, 12, 13, 14, 14, 14, 15 Steps: 1) Arrange the data from least to greatest. Identify the least and greatest values. 2) Use the 2 k rule for determining number of classes. (no less than 5) (2 k n; where n is the number of observations) 3) Find i (interval width): H = highest value L = lowest value i = H L k (you may round up) 4) Construct a frequency distribution table: Class Tally Relative Frequency 5) Sketch the histogram. Include all labels! 9

10 AP Statistics Chapter 1 Notes 1.3 Describing Quantitative Data with Numbers Measuring Center: The Mean x = sum of observations n = x 1+x 2 + +x n n or Facts about the mean: x is actually the sample mean or mean of a sample. The population mean is denoted by the Greek letter μ (mu). In general samples use letters from the Latin alphabet while population uses letters from the Greek alphabet. The mean is sensitive to extreme values (outliers). It is not a measure of center. Mean is sometimes called the of the data. (also, the fair value) Examples: 1) A survey of 14 high school students showed that, on Monday, the number of texts sent was: Find the mean number of texts the students sent on Monday ) Jed s grades in Statistics for the first 7 exams are: 88, 95, 73, 65, 99, 85, 91. What does he need to score on his 8 th exam to have an average of at least 85? Can Jed get an A in the class if only 8 exams are taken into account? Explain 3) A movie studio released 10 films in one year, earning a mean gross revenue of $12 million per film. The next year, the studio released 8 films with a mean gross of 26.5 million per film. What is the mean gross revenue for the studio over these two years? 10

11 Facts about the median (M): It s the of the distribution. Half of the observations are smaller and half are larger. If the number of observations is odd, the median M is the observation in the ordered list. If the number of observations is even, the median M is the of the two center observations in the ordered list. The median is a measure of center. The median is sometimes called the value of the data. Mean vs. Median: Mean and median measure center in different ways. Mean is not resistant to outliers, the median is resistant to outliers. The mean and median of a roughly distribution are close together. If the distribution is exactly symmetric, then mean and median are the same. In a skewed distribution, the mean is usually farther out in the than the median. Which measure of center is better? Depends on the situation. Use mean if no outliers are present, use median if outliers are present. Quartiles: Alternate Example: McDonald s Beef Sandwiches Finding the median, Q 1, and Q 3 when n is odd. Here are the amounts of fat (in grams) for McDonald s beef sandwiches, in order: Measuring Spread: The Interquartile Range (IQR): IQR = Q 3 Q 1 11

12 ** Use IQR when median is your measure of center Outliers: An observation is an outlier if it falls more than 1.5 x IQR above the third quartile of below the first quartile. Outlier = 1.5 x IQR McDonald s Chicken Sandwiches: Determine whether the Premium Crispy Chicken Club Sandwich with 28 grams of fat is an outlier. Here are the 14 amounts of fat in order: BOXPLOTS, also known as a box-and-whiskers plot, is a graphical data summary based on measures of position. It is useful for identifying and the of the distribution. It also shows the of the data set. Example: Make a boxplot for the car sales data below: Cars sold per day over a two-week period: 14, 9, 23, 7, 11, 23, 17, 11, 3, 24, 21, 2, 20, 20 Describe the resulting boxplot. Step 1: Arrange data in order. Step 2: Find the Step 3: Find the first quartile ( ) Step 4: Find the third quartile ( ) Step 5: Check for Step 6: Draw a horizontal number line and. Step 7: Complete the Boxplot. The Five-Number Summary:,,,, A boxplot is a visual representation of the five-number summary. Boxplots don t show modes of a distribution but they do identify two measures of spread ( ). 12

13 Measuring Spread: The Standard Deviation - The most common numerical description of a distribution involves the mean (to measure center) and the standard deviation (to measure spread). - The standard deviation measures the average distance of the observations from their mean. ( ) - The more spread out about the mean observations are, the greater the standard deviation. - Standard deviation has the same units of measurement as the original observations. - Standard deviation is not. Outliers can greatly affect it. - The average squared distance is called the (standard deviation) 2 - Symbols for standard deviation: S x = standard deviation of a sample (divide by n-1) σ x = standard deviation of a population (divide by n) Example: Foot Lengths Here are the foot lengths (in centimeters) for a random sample of 7 fourteen-year-olds from the United Kingdom. 25, 22, 20, 25, 24, 24, 28 Find and interpret the standard deviation: x xi x (xi x ) Sum = Sum = ***Choosing Appropriate Measures of Center and Spread: PLOT YOUR DATA A GRAPH GIVES THE BEST OVERALL PICTURE OF A DISTRIBUTION. If distribution is skewed or strong outliers are present USE: If distribution is fairly symmetric and no outliers are present USE: Numerical measures of center and spread report specific facts about a distribution, but they do not describe its entire shape. 13

14 Example: For their final project, a group of AP Statistics students investigated their belief that females text more than males. They asked a random sample of students from their school to record the number of text messages sent and received over a two-day period. Here are their data: Males Females What conclusions should students draw? Give appropriate evidence to support your answer. 14