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+ Chapter 1: Exploring Data Section 1.3 Describing Quantitative Data with Numbers The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE

+ Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1Analyzing Categorical Data 1.2Displaying Quantitative Data with Graphs 1.3Describing Quantitative Data with Numbers

+ Section 1.3 Describing Quantitative Data with Numbers After this section, you should be able to… MEASURE center with the mean and median MEASURE spread with standard deviation and interquartile range IDENTIFY outliers CONSTRUCT a boxplot using the five-number summary CALCULATE numerical summaries with technology Learning Objectives

+ Describing Quantitative Data Measuring Center: The Mean The most common measure of center is the ordinaryarithmetic average, or mean. Definition: To find the mean (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3, …, x n, their mean is:

+ Population Parameters vs.Sample Statistics What is the difference between and ? Population : A population is any entire collection of people, animals, plants or things from which we may collect data. It is the entire groupwe are interested in, which we wish to describe or draw conclusionsabout. Examples: “ all women in the Bay Area between the ages of 18 and 54. ” “ House cats raised as pets in American households. ” “ The light bulbs we are manufacturing in this plant. ” Sample: Asample is a group of units selected from a larger group (the population). By studying the sample it is hoped to draw validconclusions about the larger group. Examples: “ 500 women in the Bay Area between the ages of 18 and 54. ” “ 50 house cats drawn from households in San Francisco, CA. ” “ 10 light bulbs manufactured at this plant in the last year. ” Parameter : A parameter is a value, usually unknown (and which therefore has to be estimated), used to represent a certain populationcharacteristic. Statistic : A statistic is a quantity that is calculated from a sample of data. It is used to give information about unknown values in thecorresponding population.

+ Population Parameters vs.Sample Statistics What is the difference between and ? refers to the mean of a sample. Most of the time, thedata we ’ ll encounter can be thought of as a sample from some larger population.When we need to refer to the mean of the population,we ’ ll use the symbol

+ Resistant Measure What is a resistant measure? A resistant measure is one that is not influenced by extremevalues in a data set. Because the mean can not resist the influence of extreme values(or outliers), we say that the mean is NOT a resistant measure. Example: Class Test ScoresCalculate the mean of each data set.71, 78, 82, 84, 85, 89, 92, 10041, 78, 82, 84, 85, 89, 92, 100

+ Describing Quantitative Data Measuring Center: The Median Another common measure of center is the median. In section 1.2, we learned that the median describes themidpoint of a distribution. Definition: The median M 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 M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list.

+ Resistant Measure Is the median a resistant measure?Example: Class Test ScoresCalculate the median of each data set.71, 78, 82, 84, 85, 89, 92, 10041, 78, 82, 84, 85, 89, 92, 100

+ Shape and Center How does the shape of a distribution affect therelationship between the mean and the median? Symmetric = mean and median are about the same Skewed Right = mean > median Skewed Left = mean < median Applet:http://bcs.whfreeman.com/tps4e/%23628644__666394__http://bcs.whfreeman.com/tps4e/%23628644__666394__

+ Comparing the Mean and the Median The mean and median measure center in different ways, andboth are useful. Don ’ t confuse the “ average ” value of a variable (the mean) with its “ typical ” value, which we might describe by the median. The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median. Comparing the Mean and the Median Describing Quantitative Data

+ Measuring Spread A measure of center alone can be misleading. A useful numerical description of a distribution requires both ameasure of center and a measure of spread. Measures of Spread: Range IQR Standard Deviation

+ Describing Quantitative Data Measuring Spread: The Range The difference between the highest value and the lowestvalue in a data set Example: Class Test ScoresFind the range of each class. 71, 78, 82, 84, 85, 89, 92, 10041, 78, 82, 84, 85, 89, 92, 100 Is the range a resistant measure?

+ Describing Quantitative Data Measuring Spread: The Interquartile Range ( IQR ) The difference between the first and third quartiles in a data set Before you can find the IQR, you need to find the quartiles Quartiles: Points that divide the data set into 4 equal pieces ***Note: there are only 3 of them! 1)Arrange the observations in increasing order and locate the median M. 2)The first quartile Q 1 is the median of the observations located to the left of the median in the ordered list. 3)The third quartile Q 3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: IQR = Q 3 – Q 1 How to Calculate the Quartiles and the Interquartile Range

+ 91219 232426 282939 4042 Describing Quantitative Data Find and Interpret the IQR Example Here is the data for the amount of fat (in grams) for McDonald’s beef sandwiches. Calculate the median and the IQR. M = 26 Q 3 = 39 Q 1 = 19 IQR= Q 3 – Q 1 = 39 - 19 = 20 Interpretation: The range of the middle half of fat for McDonald’s beef sandwiches is 20 grams.

+ Describing Quantitative Data Identifying Outliers In addition to serving as a measure of spread, theinterquartile range (IQR) is used as part of a rule of thumbfor identifying outliers. Definition: 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. Example: McDonld’s Beef Sandwiches In the data, we found Q 1 =19 grams, Q 3 =39 grams, and IQR=20 grams For these data, 1.5 x IQR = 1.5(20) = 30 Q 1 - 1.5 x IQR = 19 – 30 = -11 Q 3 + 1.5 x IQR = 39 + 30= 69 Any beef sandwich with fat content higher than 69 g or lower than -11 g would be an outlier. So, there are NOT any outliers in this data. Try the example in your notes on the chicken sandwiches!

+ The Five-Number Summary What ’ s the point? The minimum and maximum values alone tell us little about the distribution as a whole. Likewise, themedian and quartiles tell us little about the tails of adistribution. To get a quick summary of both center and spread, combine all five numbers. Describing Quantitative Data Definition: 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 Q 1 M Q 3 Maximum

+ Boxplots (Box-and-Whisker Plots) The five-number summary divides the distribution roughly intoquarters. This leads to a new way to display quantitative data,the boxplot. Draw and label a number line that includes the range of the distribution. Draw a central box from Q 1 to Q 3. Note the median M inside the box. Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers. How to Make a Boxplot Describing Quantitative Data

+ Boxplot Consider our beef sandwiches data. Construct a boxplot. Example: McDonald’s Sandwiches

+ Parallel Box Plots Parallel Box Plots comparing the fat content in McDonald ’ s beef and chicken sandwiches

+ Measuring Spread Why measure spread? We often want to know not only what the central tendency is in aset of scores or values (i.e, the mean, the median or the mode),we also want to know how bunched up or “ spread out ” the scores are.

+ Describing Quantitative Data Measuring Spread: The Standard Deviation The most commonly used measure of spread is called thestandard deviation. Standard deviation measures how far each observation is from the mean. Definition: The standard deviation s x measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. This average squared distance is called the variance.

+ Describing Quantitative Data Measuring Spread: The Standard Deviation How do you calculate standard deviation? Consider the following data on the number of pets that 9 students own. 1)Calculate the mean. 2)Calculate each deviation. deviation = observation – mean = 5 deviation: 1 - 5 = -4 deviation: 8 - 5 = 3

+ Describing Quantitative Data Measuring Spread: The Standard Deviation xixi (x i -mean)(x i -mean) 2 11 - 5 = -4(-4) 2 = 16 33 - 5 = -2(-2) 2 = 4 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 55 - 5 = 0(0) 2 = 0 77 - 5 = 2(2) 2 = 4 88 - 5 = 3(3) 2 = 9 99 - 5 = 4(4) 2 = 16 Sum= 0Sum= 52 3) Square each deviation. 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 5) Calculate the square root of the variance…this is the standard deviation. What is the variance? “average” squared deviation = 52/(9-1) = 6.5 Standard deviation = square root of variance =

+ What are the similarities and differences between range,IQR and standard deviation? Similarities: all are measures of variability Differences: resistance to outliers, using all of the data What are some properties of standard deviation? Standard deviation is never negative. Standard deviation is sensitive to outliers. A single outliercan raise the standard deviation and in turn, distort thepicture of spread. For data with approximately the same mean, the greater thespread, the greater the standard deviation. If all values of a data set are the same, the standarddeviation is zero (because each value is equal to the mean).

+ Standard Deviation Example: A random sample of 5 students was asked howmany minutes they spent doing homework the previous night.Here are there responses (in minutes): 0, 25, 30, 60, 90.Calculate and interpret the standard deviation. Standard Deviation: 34.71 Interpretation: On average, the distance between the amountof time a student does homework and the mean amount oftime students spent doing homework is 34.71 minutes.

+ Choosing Measures of Center and Spread We now have a choice between two descriptions for centerand spread Mean and Standard Deviation Median and Interquartile Range 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! Choosing Measures of Center and Spread Describing Quantitative Data

+ In this section, we learned that… A numerical summary of a distribution should report at least its center and spread. The mean and median describe the center of a distribution in different ways. The mean is the average and the median is the midpoint of the values. When you use the median to indicate the center of a distribution, describe its spread using the quartiles. The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q 3 – Q 1. Summary Section 1.3 Describing Quantitative Data with Numbers

+ In this section, we learned that… An extreme observation is an outlier if it is smaller than Q 1 –(1.5xIQR) or larger than Q 3 +(1.5xIQR). The five-number summary (min, Q 1, M, Q 3, max) provides a quick overall description of distribution and can be pictured using a boxplot. The variance and its square root, the standard deviation are common measures of spread about the mean as center. The mean and standard deviation are good descriptions for symmetric distributions without outliers. The median and IQR are a better description for skewed distributions. Summary Section 1.3 Describing Quantitative Data with Numbers

+ Looking Ahead… We’ll learn how to model distributions of data… Describing Location in a Distribution Normal Distributions In the next Chapter…

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