1 13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers in the set tend to cluster, a kind of middle number or a measure of central tendency. Three such measures are discussed in this section. Mean The mean (more properly called the arithmetic mean) of a set of data items is found by adding up all the items and then dividing the sum by the number of items. (The mean is what most people associate with the word average. ) The mean of a sample is denoted (read x bar ), while the mean of a complete population is denoted (the lower case Greek letter mu). The mean of n data items x 1, x 2,, x n, is given by the formula We use the symbol for summation, (the Greek letter sigma): x x n x x1 x2... xn Ex 1) Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mean number of siblings for the ten students. Ex 2) Researchers at the University of Maryland collected body temperature readings from a sample of adults, and eight of those temperatures are listed below (in degrees Fahrenheit). Does the mean of this sample equal 98.6, which is commonly believed to be the mean body temperature of adults? Weighted Mean x f w. f The weighted mean of n numbers x 1, x 2,, x n, that are weighted by the respective factors f 1, f 2,, f n is given by the formula above. In a common system for finding a grade-point average, an A is assigned 4 points, with 3 points for a B, 2 for C, and 1 for D. Find the grade-point average by multiplying the number of units for a course and the number assigned to each grade, and then adding these products. Finally, divide this sum by the total number of units. Ex 3) Find the grade-point average (weighted mean) for the grades below. Course Grade Points Units (credits) Math 4 (A) 5 History 3 (B) 3 Health 4 (A) 2 Art 2 (C) 2 Cosner - Math Notes - 1
2 Median Another measure of central tendency, which is not so sensitive to extreme values, is the median. This measure divides a group of numbers into two parts, with half the numbers below the median and half above it. To find the median of a group of items: 1) Rank the items. (Put in order.) 2) If the number of items is, the median is the middle item in the list. 3) If the number of items is, the median is the mean of the two middle numbers. Ex 4) Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 1, 6, 3, 3, 4, 2. Find the median number of siblings for the ten students. Position of the Median in a Frequency Distribution Position of median = Notice that this formula gives the position, and not the actual value. Ex 5) Find the median for the distribution. Value Frequency n 1 f Mode The mode of a data set is the value that occurs most frequently. When two values occur with the same greatest frequency, each one is a mode and the data set is bimodal. When more than two values occur with the same greatest frequency, each ins a mode and the data set is multimodal. When no value is repeated, we say that there is no mode. Ex 6) Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mode for the number of siblings. Cosner - Math Notes - 2
3 Ex 7) Find the mode for the distribution. Value Frequency Central Tendency from Stem-and-Leaf Displays We can calculate measures of central tendency from a stem-and-leaf display. The median and mode are easily identified when the leaves are ranked (in numerical order) on their stems. Ex 8) Given the stem and leaf plot below, calculate the three measures of central tendency. Symmetry in Data Sets The most useful way to analyze a data set often depends on whether the distribution is symmetric or nonsymmetric. In a symmetric distribution, as we move out from a central point, the pattern of frequencies is the same (or nearly so) to the left and right. In a non-symmetric distribution, the patterns to the left and right are different. A non-symmetric distribution with a tail extending out to the left, shaped like a J, is called skewed to the left. If the tail extends out to the right, the distribution is skewed to the right. Cosner - Math Notes - 3
4 13.3 Measures of Dispersion 13.3 Book HW: 4 (by hand), 12 The mean is a good indicator of the central tendency of a set of data values, but it does not give the whole story about the data. Ex 1) Below are scores on quizzes in 2 classes. Which class did better? Class 1 Class x 6.3 x 6. 3 mode = 6,8 mode = 6 median = 6 median = We cannot tell by just the measures of central tendency. So we must look at the measure of dispersion, or spread, of the data. Two of the most common measures of dispersion are the range and the standard deviation. The range of a set of data is the difference between the maximum value and minimum value. The range is easy to compute; however the range does not take into account all values. So, outliers can affect the range. The standard deviation of a set of sample values is a measure of variation of values about the mean. x x Sample standard deviation: s n 1 The standard deviation is a measure of variation of all values from the mean. Outliers can dramatically increase the standard deviation. The units of the standard deviation, s, are the same as the units of the original data values. The value of the standard deviation is usually positive. Question: When is the standard deviation not positive? 2 Ex 2) Find the mean and standard deviation of the given data (in minutes) by hand and then with calculator: a),,,, x (min) x x x x 2 Totals Cosner - Math Notes - 4
5 b) 46,,,, 54 x (min) x x x x Totals c) 5,,,, 95 x (min) x x x x Totals Ex 3) Find the sample standard deviation for the frequency distribution. Value Frequency Cosner - Math Notes - 5
6 The sample variance of a set of sample values is a measure of variation equal to the square of the standard deviation. Ex 4) In the preceding example we found that for 46,,,, 54 (minutes), the standard deviation was about 2.8 minutes. Find the variance of that same example. Central tendency and dispersion are different independent aspects of a set of data. Which one is more critical can depend on the situation. For example, suppose muffins sell by the basket. If you want the most muffins possible per dollar spent, you would look for the basket with the highest average weight per muffin. Or, if the muffins are to be served on a tray where presentation is important, you would look for a basket with uniform-sized muffins that is a basket with the lowest weight dispersion. So the more desirable the basket depends on your objective. Ex 5) Two companies, A and B, sell 12 ounce jars of instant coffee. Five jars of each were randomly selected from markets, and the contents were carefully weighed, shown below: A B a) Find which company provides more coffee in their jars. b) Find which company fills its jars more consistently. It is clear that a larger dispersion value means more spread than a smaller one. But it is difficult to say exactly what a single dispersion value says about a data set. Cosner - Math Notes - 6
7 13.4 Measures of Position 13.4 Book HW: 5, 6, 7, 28, 34 In some cases we are interested in certain individual items in the data set, rather than in the set as a whole. We need a way of measuring how an item fits into the collection, how it compares to other items in the collection, or even how it compares to another item in another collection. There are several common ways of creating such measures and they are usually called measures of position. If x is a data item in a sample with mean x and standard deviation s, then the z-score of x is given by z = A z-value (or standardized value), is the number of standard deviations that a given value x is above or below the mean. (Round z to two decimal places.) It is found using the following: Ex 1) Two students, who take different history classes, had exams on the same day. Jen s score was 83 while Joy s score was 78. Which student did relatively better, given the class data shown below? Jen Joy Class mean Class standard deviation 4 5 Ex 2) Stanford Binet IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein reportedly had an IQ of 160. a) What is the difference between Einstein s IQ and the mean? b) Convert Einstein s IQ score to a z-score. c) How many standard deviations is that (the difference in part a)? d) If we consider usual IQ scores to be those that covert to z scores between -2 and 2, is Einstein s IQ usual or unusual? Percentiles A percentile is a measure of location which divides the data set into 100 groups with 1% of the values in each group. We denote percentiles as P1, P2, P3,..., P 99. If we talk about P 35, then we mean the 35 th percentile and this means that about 35% of the data values will lie below this value. If we talk about P, then we mean the th percentile and this means that % of the data values lie below this value the median. Additionally, just as when finding the median, when we find percentiles we must order the data set first. Cosner - Math Notes - 7
8 There are two ways that we want to talk about percentiles: 1. We may want to know what percentile corresponds to a known data value; or 2. We may want to know what data value corresponds to a particular percentile. To find the percentile corresponding to a known data value we compute number of values less than x percentile of value x = 100 total number of data values (Round to the nearest whole number) Ex 3) In a particular class of thirty math students, they took an exam over Chapter 3. The exam scores are as follows: a) What percentile corresponds to an exam score of 73? b) What percentile corresponds to an exam score of 45? To find the data value corresponding to a stated percentile we must define a few notations: n = total number of values in the data set k = percentile being used L = location of the desired data value in the ORDERED data set th P = k percentile k First, we calculate the value of the location or position, L, as follows: k L n 100 Now, there are two options for L: 1. It is NOT a whole number and then we round UP to the next whole number and locate the data value that occupies that position; OR 2. It is a whole number and then we find the average of the Lth and (L + 1)st numbers in the ordered data set. Cosner - Math Notes - 8
9 Ex 4) In a particular class of thirty math students, they took an exam over Chapter 3. The exam scores are as follows: Find the 23 rd percentile for the Chapter 3 exam scores. Find the 70 th percentile for the Chapter 3 exam scores. Quartiles Recall that there are 99 percentiles that divide the data set up into 100 groups. There are 3 quartiles that divide the data set up into 4 groups -- Q1, Q2, and Q 3. Quartiles are measures of location, just as percentiles, but each group contains about 25% of the data. How can we connect quartiles back to percentiles? Q P 1 25 Q P median Q 2 P 3 75 So, when we find quartiles, really we aren t finding anything new just thinking back to the corresponding percentile. Ex 5) In a particular class of thirty math students, they took an exam over Chapter 3. The exam scores are as follows: Find the quartiles for the data set. Cosner - Math Notes - 9
10 The interquartile range (IQR) is the difference between the upper ( Q 3 ) and lower ( Q 1 ) quartiles. Approximately half the data values fall within the interquartile range. IQR Q3 Q1 The semi-interquartile range is the IQR divided by 2. The midquartile is the sum of the upper and lower quartiles divide by 2. Semi-interquartile = midquartile = Q3 Q1 2 Q3 Q1 2 An outlier is a value that that falls far from what would be considered normal data values. We will define outliers in terms of the interquartile range (IQR). A data value will qualify as an outlier if either of the following conditions are met: o The data value is larger than Q3 1.5 IQR ; or o The data value is smaller than Q1 1.5 IQR. We use a diagram called a boxplot visually display the extreme values (minimum and maximum), the quartiles (lower, median, upper) and the IQR over a number line. We draw a box that shows the IQR with a line through the box at the median. Then we draw in whiskers that extend from the box out to the extreme values. The 5 number summary is just an ordered listing of the important values that are used in the box plot in parentheses: Ex 6) Find the 5 number summary and construct a box plot for the Math exam scores. (minimum, Q 1, Q 2, Q 3, maximum) Deciles are the nine values (denoted D 1, D 2,, D 9 ) along the scale that divide a data set into ten (approximately) equal parts, and quartiles are the three values (Q 1, Q 2, Q 3 ) that divide the data set into four (approximately) equal parts. Ex 7) The following are test scores (out of 100) for a particular math class. Find the sixth decile Cosner - Math Notes - 10
11 The Box Plot A box plot, or box-and-whisker plot, involves the median (a measure of central tendency), the range (a measure of dispersion), and the first and third quartiles (measures of position), all incorporated into a simple visual display. For a given set of data, a box plot, or box-and-whisker plot, consists of a rectangular box positioned above a numerical scale, extending from Q 1 to Q 3, with the value of Q 2 (the median) indicated within the box, and with whiskers (line segments) extending to the left and right from the box out to the minimum and maximum data items. Ex 8) Construct a box plot for the weekly study times data shown below Ex 9) Using the data set below to create a boxplot for the data set Minimum: Lower Quartile: Median: Upper Quartile: Maximum: IQR: Cosner - Math Notes - 11
12 13.5 The Normal Distribution 13.5 Book HW: 46,48, 52, 54, 60b A random variable that can take on only certain fixed values is called a discrete random variable. A variable whose values are not restricted in this way is a continuous random variable. A normal curve is a symmetric, bell-shaped curve. Any random variable whose graph has this characteristic shape is said to have a normal distribution. On a normal curve, if the quantity shown on the horizontal axis is the number of standard deviations from the mean, rather than values of the random variable itself, then we call the curve the standard normal curve. Properties of Normal Curves The graph of a normal curve is bell-shaped and symmetric about a vertical line through its center. The mean, median, and mode of a normal curve are all equal and occur at the center of the distribution. Empirical (or ) Rule for Data with a Bell-Shaped Distribution This rule states that for data sets having a distribution that is approximately bell-shaped, the following properties apply. About % of all values fall within standard deviation of the mean. About % of all values fall within standard deviation of the mean. About % of all values fall within standard deviation of the mean. Ex 1) Suppose that 280 sociology students take an exam and that the distribution of their scores can be treated as normal. Find the number of scores falling within 2 standard deviations of the mean. Ex 2) IQ scores have a bell-shaped distribution with a mean of 100 and variance of 225. What percentage of IQ scores are between 70 and 130? Cosner - Math Notes - 12
13 A Table of Standard Normal Curve Areas To answer questions that involve regions other than 1, 2, or 3 standard deviations of the mean we can refer to one of our tools, the calculator. The general syntax of the normalcdf command is: normalcdf( start, end, mean, std_dev) 2 nd VARS (DISTR), 2: normalcdf( The TI-83/84 will return the percentage (as a decimal number from 0 to 1) of data points x in a normal distribution with the given mean and standard deviation which lie between x = start and x = end. The standard normal distribution is a normal probability distribution with 1) 0 2) 1 3) Total area under its density curve is equal to 1. Ex 3) Use the calculator to find the percent of all scores that lie between the mean and the following values. a) 1.5 standard deviation above the mean b) 2.62 standard deviations below the mean Ex 4) Find the total area indicated in the region in color below. 0 z z Cosner - Math Notes - 13
14 Ex 5) Find the following using the table by first drawing a sketch: a) P(z < 1.58) b) P(z < -1.45) c) P(-2.3 < z < -1.45) Interpreting Normal Curve Areas In a standard normal curve, the following three quantities are equivalent. 1) Percentage (of total items that lie in an interval) 2) Probability (of a randomly chosen item lying in an interval) 3) Area (under the normal curve along an interval) Ex 6) The volumes of soda in bottles from a small company are distributed normally with mean 12 ounces and standard deviation.15 ounce. If 1 bottle is randomly selected, what is the probability that it will have more than ounces? Cosner - Math Notes - 14
15 Ex 7) The volume of liquid in a bottle of eye drop medicine is normally distributed with a mean of 30.5 ml and standard deviation of 2.5 ml. Calculate the following probabilities of randomly selecting a bottle: a) with less than 32 ml of eye drop medicine. b) with the amount of eye drop medicine between 26.5 and 34 ml. Ex 8) Assuming a normal distribution, find the z-value meeting the condition that 39% of the area is to the right of z. Ex 9) The shrinkage in length of a certain brand of blue jeans is normally distributed with a mean of 1.35 inches and standard deviation of 0.25 inch. What percent of this brand of jeans will shrink between 1 and 2 inches? Cosner - Math Notes - 15
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290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since