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1.3 Describing Quantitative Data with Numbers Pages 48-73 Objectives SWBAT: 1) Calculate measures of center (mean, median). 2) Calculate and interpret measures of spread (range, IQR, standard deviation). 3) Choose the most appropriate measure of center and spread in a given setting. 4) Identify outliers using the 1.5 X IQR rule. 5) Make and interpret boxplots of quantitative data. 6) Use appropriate graphs and numerical summaries to compare distributions of quantitative variables. • Measuring Center: The Mean – The most common measure of center is the ordinary arithmetic average, or mean. Definition: To find the mean x (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x1, x2, x3, …, xn, their mean is: sum of observations x1 x 2 ... x n x n n In mathematics, the capital Greek letter Σis short for “add them all up.” Therefore, the formula for the mean can be written in more compact notation: x x n i What is a resistant measure? Is the mean a resistant measure of center? • A resistant measure is a measure that can resist the influence of extreme observations. • Think about if we were going to calculate the mean salary for students in this classroom. • Let’s say Adam Sandler finds out he is one class short of graduating high school, and that class happens to be AP Statistics. He moves to Lyndhurst and transfers into this class. What effect would his salary have on the mean? • What type of effect would it have on the median? • Because the mean cannot resist the influence of extreme observations, we say that it is not a resistant measure of center. However, median is a resistant measure of center. How can you estimate the mean of a histogram or a dotplot? • The mean is the balancing point of the distribution, so the mean of the deviations from the mean will add to 0 and balance there. • Where does it look like the balancing point of this distribution will be? Measuring Center: The Median Another common measure of center is the median. The median describes the midpoint of a distribution. The median is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1. Arrange all observations from smallest to largest. 2. If the number of observations n is odd, the median is the center observation in the ordered list. 3. If the number of observations n is even, the median is the average of the two center observations in the ordered list. As previously stated, the median is a resistant measure of center. How does the shape of a distribution affect the relationship between mean and median? There is a connection between the shape of a distribution and the relationship between the mean and median of the distribution. • When a distribution is symmetric, the mean and median will be approximately the same. • When a distribution is skewed right, the mean will be greater than the median. • When a distribution is skewed left, the mean will be smaller than the median. • This distribution of stolen bases is skewed right, with a median of 5, as noted on the histogram. • It does not seem plausible that the balancing point (mean) is also 5. Because the distribution is stretched out to the right, the mean must be greater than 5. Think of all the extremely values that will pull the mean up. • Two common measures of spread (variability) are range and IQR. What is range? Is it a resistant measure of spread? • The range of a distribution is the distance between the minimum value and the maximum value. • Do you think it is a resistant measure of spread? Let’s go back to our Adam Sandler example. • Range can be a bit deceptive if there is an unusually high or unusually low value in a distribution. It is not a resistant measure of spread. What are quartiles? How do you find them? • Quartiles are the values that divide a distribution into four groups of roughly the same size. • Find the quartiles: 4 6 8 12 20 22 27 How To Calculate The Quartiles And The IQR: To calculate the quartiles: 1.Arrange the observations in increasing order and locate the median. 2.The first quartile Q1 is the median of the observations located to the left of the median in the ordered list. 3.The third quartile Q3 is the median of the observations located to the right of the median in the ordered list. What is the interquartile range (IQR)? Is the IQR a resistant measure of spread? • The interquartile range (IQR) is a single number that measures the range of the middle half of the distribution, ignoring the values in the lowest quarter of the distribution and the values in the highest quarter of the distribution The interquartile range (IQR) is defined as: IQR = Q3 – Q1 • Since IQR essentially discards the lowest and highest 25% of the distribution, any outliers would have a minimal affect on IQR. As a result, we can state that IQR is a resistant measure of spread. Find and Interpret the IQR Travel times for 20 New Yorkers: 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45 5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85 Q1 = 15 Median = 22.5 Q3= 42.5 IQR = Q3 – Q1 = 42.5 – 15 = 27.5 minutes Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes. Here are data on the amount of fat (in grams) in 9 different McDonald’s fish and chicken sandwiches. Calculate the median and the IQR. Median=19 What is an outlier? How do you identify them? Are there any outliers in the chicken/fish distribution? In addition to serving as a measure of spread, the interquartile range (IQR) is used as part of a rule of thumb for identifying outliers. The 1.5 x IQR Rule for Outliers Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile. Since no values fall below the boundary of 0.75 or above the boundary of 38.75, the distribution contains no outliers. Are there any outliers in the beef distribution? No values fall below 9, so there are no small outliers. 43 falls above 41, so the Double Quarter Pounder with Cheese is an outlier. The Five-Number Summary The minimum and maximum values alone tell us little about the distribution as a whole. Likewise, the median and quartiles tell us little about the tails of a distribution. To get a quick summary of both center and spread, combine all five numbers. The five-number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. Minimum Q1 Median Q3 Maximum Boxplots (Box-and-Whisker Plots) The five-number summary divides the distribution roughly into quarters. This leads to a new way to display quantitative data, the boxplot. How To Make A Boxplot: • A central box is drawn from the first quartile (Q1) to the third quartile (Q3). • A line in the box marks the median. • Lines (called whiskers) extend from the box out to the smallest and largest observations that are not outliers. • Outliers are marked with a special symbol such as an asterisk (*). Construct a Boxplot Consider our New York travel time data: 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45 5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85 Min=5 Q1 = 15 Median = 22.5 Q3= 42.5 Max=85 Recall, this is an outlier by the 1.5 x IQR rule • Note: When describing the shape of a distribution using a boxplot, we cannot see peaks, so we can only describe it using symmetric, skewed left, or skewed right. • To make a boxplot with the TI, see the technology corner videos in Chapter 1 on the book website, or page 59 of the textbook. • In the distribution above, how far are the values from the mean, on average? • The concept of mean absolute deviation is similar to standard deviation. Measuring Spread: The Standard Deviation The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Consider the following data on the number of pets owned by a group of 9 children. 1) Calculate the mean. 2) Calculate each deviation. deviation = observation – mean deviation: 1 - 5 = - 4 deviation: 8 - 5 = 3 x=5 Measuring Spread: The Standard Deviation (xi-mean)2 xi (xi-mean) 1 1 - 5 = -4 (-4)2 = 16 3 3 - 5 = -2 (-2)2 = 4 3) Square each deviation. 4 4 - 5 = -1 (-1)2 = 1 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 4 4 - 5 = -1 (-1)2 = 1 4 4 - 5 = -1 (-1)2 = 1 5 5-5=0 (0)2 = 0 7 7-5=2 (2)2 = 4 8 8-5=3 (3)2 = 9 9 9-5=4 (4)2 = 16 5) Calculate the square root of the variance…this is the standard deviation. Sum=? “average” squared deviation = 52/(9-1) = 6.5 Standard deviation = square root of variance = Sum=? This is the variance. 6.5 = 2.55 • A few words on standard deviation: – It measures the average distance of observations from their mean. – The formula for standard deviation and variance are on the formula sheet. However, the calculations can be done right on the 84. – For now, just know that variance is standard deviation squared. What are some similarities and differences between range, IQR, and standard deviation? • Similarity: All three measure variability. • Differences: – only IQR is resistant to outliers – only standard deviation uses all the data What are some properties of standard deviation? • SD measures the spread about the mean and should be used only when the mean is chosen as the center (if median is chosen, use IQR). • SD is always greater than or equal to 0. [A SD of 0 would mean all observations are the same value.] • SD has the same units of measurement as the original observations. • SD is not resistant to outliers. A few outliers can make SD very large. A random sample of 5 students was asked how many minutes they spent doing HW the previous night. Here are their responses (in minutes): 0, 25, 30, 60, 90. Calculate and interpret the standard deviation. Choosing Measures of Center and Spread We now have a choice between two descriptions for center and spread • Mean and Standard Deviation • Median and Interquartile Range Choosing Measures of Center and Spread •The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. •Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. •NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA! Organizing a Statistical Problem As you learn more about statistics, you will be asked to solve more complex problems. Here is a four-step process you can follow. How to Organize a Statistical Problem: A Four-Step Process •State: What’s the question that you’re trying to answer? •Plan: How will you go about answering the question? What statistical techniques does this problem call for? •Do: Make graphs and carry out needed calculations. •Conclude: Give your conclusion in the setting of the real-world problem.