Elementary Statistics - PDF Free Download

Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q: Can we model the data? Q: How do we know if we have a good model? Q: Is our data affected by other variables? Definitions Individuals : Objects described by a set of data. Individuals may be people, but they may also be animals or things. Variable : Any characteristic of an individual. A variable can take on different values for different individuals. Categorical and Quantitative Variables Categorical variable : Places an individual into one or several categories. Quantitative variable : Takes numerical values for which arithmetic operations make sense. Distribution : Tells what values the data takes and how often it takes these values. ١

Dataset, Individuals and Variables Variables Company Observation Stock Annual Earn/ Exchange Sales($M) Sh.($) Dataram AMEX 73.10 0.86 EnergySouth OTC 74.00 1.67 Keystone NYSE 365.70 0.86 LandCare NYSE 111.40 0.33 Psychemedics AMEX 17.60 0.13 Individuals Data Set Datum 1.1: Describing Distributions with Graphs Two Basic Strategies : 1) Begin by examining each variable by itself. Then move on to study the relationships among variables. 2) Begin with a graph or graphs. Then add numerical summaries of specific aspects of data. Different types of graphs : Bar graph, Pie chart, Stemplot, back-to-back Stemplot, Histogram, Time plot Categorical Variables Pie charts Bar graphs ٢

Bar Graphs Bar graph - A graph which displays the data using heights of bars to represent the counts of the variables. Example : Consider the following grade distribution : Grade Count A B C D Other 6 12 15 9 3 How could we display the data using a bar graph? Bar Graphs Grade Count A B C D Other 6 12 15 9 3 15 12 9 6 3 A B C D F Pie Charts Pie Chart : 1) A chart which represents the data using percentages. 2) Break up a circle (pie) into the respected percentages. ٣

Grade Count Percent Pie Charts A B C D Other 6 12 15 9 3 13 27 33 20 7 B A D F C Quantitative Variables Stem-and-Leaf (Stemplot) Histogram Describing Distributions When describing a distribution we describe four things: (1) the shape of the distribution (2) the center and the spread of the distribution (3) any unusual features in the distribution ٤

Stemplot Example: Here are the grades Max achieved while in school his first two years. Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66 Steps 1 and 2 : 4 5 6 7 8 9 10 4 5 Step 3 : 6 7 8 9 10 5 8 6 2 8 1 0 7 3 0 7 6 3 6 4 1 2 2 3 8 Stemplot Example: Here are the grades Max achieved while in school his first two years. Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66 Steps 1 and 2 : 4 5 6 7 8 9 10 4 5 Step 3 : 6 7 8 9 10 5 8 6 2 2 0 0 2 3 1 6 7 3 3 1 4 7 6 8 8 Back-To-Back Stemplot This is a stemplot which allows you to see and compare the distribution of two related data sets Example : Here are the grades Lulu received during her first two years at college : Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95, 87, 76, 93, 82, 66, 75 To make a Back-To-Back Stemplot, you make the stem, and the stems going off to the right and the left. You want the smaller values closer to the stem. ٥

Back-To-Back Stemplot Lulu s Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95, 87, 76, 93, 82, 66, 75 Max s Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66 5 6 6 5 6 3 1 8 8 7 2 7 2 6 4 3 5 3 0 2 4 5 6 7 8 9 10 5 8 6 2 2 0 0 2 3 1 6 7 3 3 1 4 7 6 8 8 Back-To-Back Stemplot Lulu s Grades: 66, 77, 78, 84, 92, 90, 86, 78, 71, 93, 82, 55, 73, 95, 87, 76, 93, 82, 66, 75 Max s Grades: 88, 72, 91, 83, 77, 90, 45, 83, 94, 91, 86, 77, 82, 100, 58, 76, 83, 88, 72, 66 5 6 6 8 8 7 6 5 3 1 7 6 4 2 2 5 3 3 20 4 5 6 7 8 9 10 5 8 6 2 2 0 0 2 3 1 6 7 3 3 1 4 7 6 8 8 Splitting Stems If you have a large data set (leaves), then sometimes a stemplot will not work very well. For instance, if you have a large amount of leaves, and only a few stems, you might want to split the stems. Example : Consider the following test scores : 71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84, 85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97 Normally we would set up the stems as follows : 7 8 9 ٦

Splitting Stems If you have a large data set (leaves), then sometimes a stemplot will not work very well. For instance, if you have a large amount of leaves, and only a few stems, you might want to split the stems. Example : Consider the following test scores : 71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84, 85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97 Normally we would set up the stems as follows : 7 8 9 1, 1, 2, 4, 5, 5, 6, 7, 9 0, 1, 1, 2, 3, 3, 3, 3, 4, 5, 5, 8, 9 0, 0, 0, 1, 3, 5, 6, 7 Splitting Stems If you have a large data set (leaves), then sometimes a stemplot will not work very well. For instance, if you have a large amount of leaves, and only a few stems, you might want to split the stems. Example : Consider the following test scores : 71, 71, 72, 74, 75, 75, 75, 76, 77, 79, 80, 81, 81, 82, 83, 83, 83, 83, 84, 85, 85, 88, 89, 90, 90, 90, 91, 93, 95, 96, 97 However, we could set up the stems as follows : This stem gets scores 70-74 This stem gets scores 75-79 7 7 8 8 9 9 1 1 2 4 5 55679 0 11233334 5 5 8 9 0 0013 5 6 7 Stem-and-Leaf Plot CAN: 1. Determine the center of the distribution. 2. Determine the range or spread of the data. 3. Determine the shape of the distribution. 4. Determine any range of values not represented. Determine if there is a concentration of data. Determine if there are any outliers (extreme values). ٧

Advantages Graphically Display the distribution of the data Retain the actual data Easy to construct Make sorting of the data easy Disadvantages Not very effective for large data sets (would take a long time to construct). Choice of the stems depends on the data type and data range. Different choices for the stems can cause different looking distributions. Histograms A histogram breaks the range of variables up into (equal) intervals, and displays only the count or percent of the observations which fall into the particular intervals. Notes: You can choose the intervals (usually equal) Slower to construct than stemplots Histograms do not display the individual observations In case a score falls on an interval point, you must decide in advance which interval in which the point will go. ٨

Steps to drawing a histogram : Histograms 1) Divide the range into classes of equal width. 2) Count the number of observations in each class. These are called frequencies. 3) Draw the histogram. Histograms 1a. Determine the number of class intervals to use. One rule is to calculate the square root of the sample size, and round up. Example: If the sample consists of 210 subjects,then the square root of 210 is 14.5, which is rounded up to 15. Histograms 1b. Determine the range of the data by subtracting the smallest observation from the largest observation. Example: If the smallest observation is 202 and the largest observation is 496, then the range is 496-202 = 294. ٩

Histograms 1c. Divide the range by the number of class intervals and round to a convenient number. This will be the equal class width. Example: Range = 294 Number of intervals = 15 294/15 = 19.6 round to 20 Use a class width of 20 for each interval. Histograms 1d. The lower limit of the first interval should be a multiple of the class width and should be chosen such that the smallest observation is contained in the first interval. Example: Multiples of 20 are: 0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260,.. The smallest observation is 202. Choose the lower limit of the first interval to be 200. Histograms The rest of the intervals are obtained by adding the class width to this first lower limit value. Example: 200-220 220-240 240-260... 480-500 ١٠

Histograms 2a. Count the number of observations falling in each interval. These counts are referred to as class frequencies (or just frequencies). As a rule, an observation that falls on the boundary of two intervals should be placed in the second interval, not the first. Example: 220 goes in the interval 220-240, not 200-220 Histograms 2b. Determine the relative frequency for each class interval by dividing the class frequency by the total number of observations and multiplying by 100. The relative frequencies are the percentages of the observations in each interval. Example: If the frequency in the 200-220 class is 4 and the number of observations (n) is 210, then the relative frequency is (4/210)*100 = 1.9% Histograms Display the class intervals, class frequencies, and relative frequencies in a frequency table. Class Intervals 200-220 220-240 240-260 260-280... Frequencies 16 5 18 4... % 25.5% 9.5% 10.5% 9.0% ١١

Histograms 3. Construct the histogram. On the horizontal axis, mark and label the class intervals. On the vertical axis, mark and label the class frequencies(to create a frequency histogram) or the relative frequencies (to create a relative frequency histogram). Over each class interval, draw a rectangle whose height equals the correct frequency or relative frequency. Histograms Example : Suppose the final breakdown in grades looks like this : Grade Amount Percent Frequency Table 90-100 8 20 80-90 10 25 70-80 10 25 60-70 8 20 50-60 4 10 4 8 10 10 8 50 60 70 80 90 100 Histograms Example : Suppose the final breakdown in grades looks like this : Grade Amount Percent 90-100 8 20 80-90 10 25 70-80 10 25 60-70 8 20 50-60 4 10 10% 20% 25% 25% 20% 50 60 70 80 90 100 ١٢

Population Density Curves Smoothed version of a relative frequency histogram, such that the area under the curve represents relative frequency and the total area is 1. Symmetric (Normal) Curve Bell-shaped curve. Most commonly used type of distribution. Basis for many statistical inference procedures. Symmetric (Normal) Curve ١٣

Skewed Left Distribution General bell-shape, with a long tail to the left Skewed Left Distribution Skewed Right Distribution General bell-shape, with a long tail to the right. ١٤

Skewed Right Distribution Bimodal Distribution A distribution with two significant peaks. Bimodal Distribution ١٥

Trimodal Distribution A distribution with three significant peaks. Trimodal Distribution Describing a Distribution 1. Shape 2. Center and Spread 3. Unusual features (outliers, gaps, high concentrations of data) ١٦

Example 1 Grades Distribution 0 0 where 3 7 = 37 1 2 3 7 4 3,4,8 5 6 8 7 7,9,7,6,5,4,8 8 2,0,8,0,5,4,5,5,3,3,6,4,8,7,6,3,9 9 2,1,4,5,1,2,3 Skewed left, center around 80, range from 0 to 95, at least one outlier Example 2 0 95 1 27,59 2 48,68,56 3 59,38 Bimodal, 4 50,52 center around 500, 5 03 6 80 ranges from 95 to 936, 7 79,43 no outliers 8 79,42,05,94 9 01,36 where 1 27 = 127 Example 3 0. 3, 0 where 1 2 = 12 0 * 1. 2, 2 1 * 7, 8, 8, 8, 9 2. 2, 1, 1, 1 2 * 7, 7, 9, 6, 8, 9, 6 3. 0, 2, 3, 2, 1, 2, 3, 1, 2 3 * 5, 8, 6, 5, 8 4. 2, 1, 3 4* 7, 5 5. 1 5* Symmetric, center around 30, range from 0 to 51 missing values at 0* ١٧

Example 4: 1988 State Data 1 11 7,2,8,9,3,8,5,9,3,8,7,2,9 12 6,9 13 6,9,3,8,7,7 14 5,5,3,5,5,1,8,3,0 15 9,2,7,6,8,6,5 16 7,5,6,7 17 9 18 3,4,5,1 19 8 20 21 0 22 1 23 24 25 Skewed Right, Bimodal???? Center between 15,000 and 16,000 Range from 11,100 to 23,100 Two outliers Example 5: 1991 State Data 13 3 14 8,8,5,2 15 6,4,5,1,1,8,4 16 4,0,6,1,4,3 17 4,5,5,8,9,9,6,3,7,1 18 9,5,7,8,0 19 4,1,2,1,4 20 3,8,0 21 9,0,3,0 22 1,9,5 23 24 25 9,4 Roughly symmetric, center around 18,000, range from 13,300 to 25,900, two possible outliers, Example 6 Relative Frequency 20% 16% 12% Roughly symmetric, (bimodal), center around 35, range from 12 to 60, no outliers, 8% 4% 0% 12 18 24 30 36 42 48 54 60 Test Scores ١٨

Time Plot A Time Plot is a graph with two axis. One axis represents time,and the other axis represents the variable being measured. Variable Time Time Plot Example : The following are homerun totals for a certain baseball player the last 10 years : Year HR 89 90 91 92 93 94 95 96 97 98 33 39 22 42 9 9 39 52 58 70 Construct a timeplot for this data set. Time Plot Year HR 89 90 91 92 93 94 95 96 97 98 33 39 22 42 9 9 39 52 58 70 Home Run Year ١٩

Time Plot Year HR 89 90 91 92 93 94 95 96 97 98 33 39 22 42 9 9 39 52 58 70 70 60 50 40 30 20 10 89 90 91 92 93 94 95 96 97 98 What to look for in a distribution 1) Look for the overall pattern and for deviations from the pattern See if the distribution has a shape we can describe in a few words Describe the center and spread of the distribution 2) One common deviation from the overall pattern in any graph of data is an outlier, i.e., an observation that falls outside the overall pattern of the graph 1.2: Describing Distributions with Numbers A. Measures of Location (Center) Mean Median B. Measures of Spread (Variability) Quartiles (Quantiles) Variance and Standard deviation ٢٠

Measures of Location 1. Mean (Average) How to find the mean (average): 1) Add the values together 2) Divide the total by the number of observations Example: Test Scores : 56, 65, 54, 55, 57, 54, 61, 62, 60, 55, 57, 56, 57, 61, 62, 60, 49, 66, 59, 80 Step 1 : 56 + 65 + 54 + + 59 + 80 = 1186 Step 2 : 1186 / 20 = 59.3 Mean Mean To find the mean x 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 : x = x 1 + x 2 + x 3 +... + n x n Or, in more compact notation: x = x i 2. Median How to find the median M : 1) Arrange the observations in order from smallest to largest. 2) If the number of observations is odd, then the median is located at the center of the list. So, if there are n observations, then the median is located in spot (n + 1) / 2 3) If the number of observations is even, then the median is the average of the two terms in the middle spots. These are located in spots (n / 2) and (n / 2) + 1 ٢١

Median Example of finding a Median : List 1 : 2, 4, 6, 3, 5, 2, 6, 8, 10, 11, 1 Step 1: Order the list : 1, 2, 2, 3, 4, 5, 6, 6, 8, 10, 11 Step 2 : Find the middle term2 : (n+1) / 2 = (11 + 1) / 2 = 6 1, 2, 2, 3, 4, 5, 6, 6, 8, 10, 11 Median Median Example of finding a Median : List : 2, 4, 6, 3, 5, 2, 6, 8, 10, 11, 1, 12 Step 1: Order the list : 1, 2, 2, 3, 4, 5, 6, 6, 8, 10, 11, 12 Step 2 : Find the two middle terms : n / 2 = 12 / 2 = 6 (n / 2) + 1 = (12 / 2) + 1 = 7 Step 3 : Average the sixth and seventh terms : 1, 2, 2, 3, 4, 5, 6, 6, 8, 10, 11, 12 Median = (5 + 6) /2 = 5.5 In The Presence Of Outliers Q: Do outliers affect the Mean and Median? Consider the list on numbers from 1 through 9 : 1, 2, 3, 4, 5, 6, 7,8,9 The Mean is : 5 The Median is : 5 What if we put the number 100 at the end of the list : 1, 2, 3, 4, 5, 6, 7,8,9, 100 The Mean is : 14.5 The Median is : 5.5 A: Outliers affect the Mean much more than the Median! ٢٢

Distributions The mean is the point at which a histogram balances. For symmetric distributions the mean and median will be nearly the same. X M Distributions The mean is the point at which a histogram balances. For symmetric distributions the mean and median will be nearly the same. However, since the mean is influenced by outliers, for skewed distributions the mean will be pulled in the direction of the long tail while the median will be resistant to the outliers and remain in nearly the same place. Skewed Right M X ٢٣

Skewed Left X M Measures of Spread Consider the following pay distributions: Low High Center Low High Measuring Spread The simplest useful numerical description of a distribution consists of both a measure of center and a measure of spread. 1. Percentiles and Quantiles Definition: The pth percentile of a distribution is the value such that p percent of the observations fall at or below it. Example: The Median is the 50th percentile. Q: Why isn t the Mean the 50th percentile? 1, 2, 3, 4, 5, 6, 7,8,9, 100 The Mean is : 14.5 The Median is : 5.5 ٢٤

Describing Spread The Five Number Summary : 1) The Median 2) First Quartile : 25% of the observations lie below the First Quartile 3) Third Quartile : 75% of the observations lie below the third quartile 4) Lowest Individual Observation (Minimum) 5) Highest Individual Observation (Maximum) Quartiles Calculating the Quartiles : 1) Arrange the observations in increasing order and locate the Median M in the ordered list o observations. 2) The First Quartile Q1 is the median of the observations whose position in the ordered list is to the left of the location of the overall median. 3) The Third Quartile Q3 is the median of the observations whose position in the ordered list is to the right of the location of the overall median. Quartiles Example of calculating First Quartile : List of quiz scores: 10, 8, 9, 4, 6, 6, 8, 9, 2, 7 1) Order the list: 2, 4, 6, 6, 7, 8, 8, 9, 9, 10 Find the median: (7 + 8) / 2 = 7.5 2) Find all the observations whose position in the list is to the left of the median : 2, 4, 6, 6, 7, 8, 8, 9, 9, 10 Find the median of these values : 6 ٢٥

Quartiles Example of calculating Third Quartile : List of quiz scores: 10, 8, 9, 4, 6, 6, 8, 9, 2, 7, 11 1) Order the list: 2, 4, 6, 6, 7, 8, 8, 9, 9, 10, 11 Find the median: 8 2) Find all the observations whose position in the list is to the right of the median : 2, 4, 6, 6, 7, 8, 8, 9, 9, 10, 11 Find the median of these values : 9 Interquartile Range The interquartile range, IQR, is the distance between the first quartile and the third quartile. Determining Outliers Call an observation a suspected outlier if it falls more than 1.5 * IQR above the third quartile or below the first quartile. Example : Imagine we have a bunch of test scores with Q1 = 50 and Q3 = 80. The IQR = 80-50 = 30 So, 1.5 * IQR = 1.5 * 30 = 45 This means that if there are any scores above Q3 + 45 = 125 or any scores Q1-45 = 5, then these scores are suspected outliers. Boxplot Example: Low = 47, High = 98, Median = 77, Q1 = 65, Q3 = 85 A Boxplot is a graph of the five number summary. A central box spans the quartiles, with a line marking the median. Whiskers extend out from the box to the extremes. Highest Observation (98) 90 Q3 (85) Median (77) 70 Q1 (65) 50 Lowest Observation (47) 30 10 0 ٢٦

Describing Spread 2. The Standard Deviation Variance: The variance of a set of observations is an average of the deviations of the observations from the mean. Note: You divide by (n - 1) instead of n. Standard Deviation: The SD is the square root of the variance. Describing Spread The Standard Deviation Example : Test Scores : 65, 77, 83, 80, 95 1) Find the average : 80 2) Find the deviations from the mean, and their squares Obs Deviation from Mean Deviations Squared 65-15 225 77-3 9 83 3 9 80 0 0 95 15 225 Describing Spread The Standard Deviation 3) Determine the mean of the squares: (225 + 9 + 9 + 0 + 225) = 117 (5-1) 4) Determine the Standard Deviation: Variance 117 = 10.8 ٢٧

More Fancy Notation s 2 The variance of a set of observations is the average of the squares of the deviations of the observations from their mean. In symbols, the variance on n observations x 1,,... is : s 2 = 2 (x - x ) 1 + x 2 2 (x - x ) + 2 n - 1 x n... + 2 (x - x ) n or, in more compact notation : 1 s 2 2 = (x - x ) n-1 i The standard deviation s is the square root of the variance s 2 : s = 1 n-1 2 (x - x ) i Another Example of Standard Deviation Consider the following years in our past : 1792, 1666, 1362, 1614, 1460, 1867, 1439 Find the standard deviation of these years. The Mean = 1600 1792 1666 1362 1614 1460 1867 1439 x i ( x i -x) 192 66-238 14-140 267-161 2 ( x i -x) 36864 4356 56644 196 19600 71298 25921 1 s 2 2 = (x - x ) n-1 i = 1 6 ( 214879 ) = 35813.166 s = 189.2 Why Do We Square The Deviations? 1) The sum of the squared deviations of any set of observations from their mean is the smallest that the sum of squared deviations from any number can possibly be. Why use the Standard Deviation and not the Variance? 1) The standard deviation is the measure of spread for an important class of symmetric unimodal distributions called the normal distribution. 2) The standard deviation is used by the normal distribution. 3) The variance uses squared deviations, which gives a different unit from the original data. Why use n - 1? 1) The sum of the deviations is *always* zero. So, if we know n-1 of the deviations, then the last deviation can be calculated. So, only n-1 of the deviations can vary freely. These are called degrees of freedom. ٢٨

Properties of Standard Deviations 1) The standard deviation measures spread about the mean and should be used only when the mean is chosen as the measure of center. 2) s = 0 only when there is no spread. This happens only when all observations have the same value. Otherwise, s > 0. As the observations get more spread out from the mean, then s gets larger. 3) s, like the mean, is not resistant. A few outliers can make s very large. Which Measure To Use? Q: When is the mean better than median? When is the five number summary better than the standard deviation? Rules Of Thumb A1: If outliers appear, or if your distribution is skewed, then the mean could be affected, so use the median and the five number summary. A2: If the distribution is reasonably symmetric and is free of outliers, then the mean and standard deviation should be used. Changing Units Consider the following values : 30, 40, 50, 60, 70 The mean is 50 and the standard deviation is 15.8 What happens to these if we take every score, multiply it by 2 and add 10 We get these values : 70, 90, 110, 130, 150 The mean is 110 and the standard deviation is 31.6 ٢٩

Changing Units Old values : 30, 40, 50, 60, 70 mean = 50 and s = 15.8 What happens to these if we take every score, multiply it by 2 and add 10 New values : 70, 90, 110, 130, 150 mean = 110 and s = 31.6 150 130 110 90 70 50 30 150 130 110 90 70 50 30 150 130 110 90 70 50 30 Linear Transformations A linear transformation changes the original variable x into the new variable given an equation of the form : x new x new = bx + a Note: The constant a shifts all values of x either up or down by the value a. The constant b changes the size of the unit of the distribution. Effects of Linear Transformations 1) To get the new spread, multiply the old spread by b. 2) To get the new mean, multiply the old mean by b and add the constant a. 1.3: The Normal Distributions Density Curves A density curve is a curve that : 1) is always on or above the vertical axis, and 2) has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values is the relative frequency of all observations that fall in that range. ٣٠

Density Curves Normal and Skewed Curves Median Mean Mean and Median of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. ٣١

Normal Curves Normal Curves are curves which are symmetric, unimodal, and bell shaped. µ represents the mean σ represents the standard deviation Equation for the curve : y = σ 1 2π e -1 2 x - µ ( ) 2 σ Why are Normal Distributions important in stats? 1) Normal distributions are good descriptions for some distributions of real data. 2) Normal distributions are good to the results of many kinds of chance outcomes. 3) Many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions. The 68-95 - 99.7 Rule In the normal distribution with mean µ and standard deviation σ : 68 % of the observations fall within σ of the mean µ 95 % of the observations fall within 2σ of the mean µ 99.7 % of the observations fall within 3σ of the mean µ ٣٢

Normal Curve Example John collected data on the heights of women ages 18 to 24. He found that the distribution was roughly normal, with a mean of 64.5 inches and a standard deviation of 2.5 inches. Normal Curve Example John collected data on the heights of women ages 18 to 24. He found that the distribution was roughly normal, with a mean of 64.5 inches and a standard deviation of 2.5 inches. Q1 : What percentage of these women were between the heights of 62 and 67 inches? Q2 : What percentage of these women were between the heights of 59.5 and 69.5 inches? Q3 : What percentage of these women were less than 64.5 inches tall? Q4 : What percentage of these women were less than 67 inches tall? Q5 : What percentage of these women were between the heights of 57 and 69.5 inches? Other Questions Q : What percentage of these women were between the heights of 60 and 70 inches? Q : Who is considered more extraordinary, a 72 inch tall female or a 72 inch tall male? Q : Who is considered more extraordinary, a 67 inch tall female or a 72 inch tall male? Q : If you get a 26 on your ACT, and your neighbor gets a 1000 on their SAT, who did better? We can answer these questions by a normalizing technique. ٣٣

Normalizing Data If we have two unrelated data sets, and they are both roughly normal, then we can perform a linear transformation on both data sets. This transformation will allow us to compare the data sets by examining how many standard deviations above or below the mean each score is. Example : Mike has an ACT score of 26 and Carol has an SAT score of 1250. Q : Who really has the better score? A : Mike s ACT score is 1.2 standard deviations above the mean, and Carol s SAT score is 1.4 standard deviations above the mean. This means that Carol actually did better on her test than Mike! Standardizing Observations If x is an observation from a roughly symmetric distribution that has mean µ and standard deviation σ, then the standard value of x is : z = x - µ σ Note: A standardized score is often called a z-score. Example : Women s IQ s have a symmetric distribution with a mean of 97 and a standard deviation of 6. What is the standard score for a woman with an IQ of 106? z = 106-97 6 = 9 6 = 1.5 Standardizing Observations If x is an observation from a roughly symmetric distribution that has mean µ and standard deviation σ, then the standard value of x is : z = x - µ σ Note: A standardized score is often called a z-score. Example : Men s IQ s have a roughly symmetric distribution with a mean of 72 and a standard deviation of 8. What is the standard score for a man with an IQ of 66? z = 66-72 8 = -6 8 = -.75 ٣٤

Deep Thoughts 1) When we are normalizing our data set, we are really performing a linear transformation. This transformation will result in the data set still being normal. 2) If we start off with a distribution which is normal, with mean µ and standard deviation σ, (denoted by N(µ, σ) ), then after we have standardized the data set, we will have a normal distribution, with mean 0 and standard deviation 1. (Denoted by N(0, 1) ). The Standard Normal Distribution If x is an observation from a roughly symmetric distribution that has mean µ and standard deviation σ, then the standard value of x is : z = x - µ σ Note: A standardized score is often called a z-score. Example : Men s IQ s have a roughly symmetric distribution with a mean of 72 and a standard deviation of 8. What is the standard score for a man with an IQ of 66? 66-72 z = = -6 = -.75 8 8 Q: What percentage of people have a score below 66? The Standard Normal Table Table A is a table of areas under the standard normal curve. The table entry for each z value is the area under the curve to the left of z ٣٥

The Standard Normal Table Example : Imagine we have done an experiment, and we want to find what percentage of people fell under a score, namely x. We then proceed to find that the z-score for the value x is -1.10..1357 The Standard Normal Table Example : The Graduate Record Examinations (GRE) are widely used to help predict the performance of applicants to graduate schools. The range of possible sores on a GRE is 200 to 900. The psychology department at a university finds the scores of its applicants on the quantitative GRE are approximately normal with mean µ = 544 and standard deviation σ = 103. Answer the following : 1) Find the percentage of people who scored 700 or higher on the test. 2) Find the percentage of people who scored below 500 on the test. 3) Find the percentage of people who scored between 500 and 800 on the test. 1) Find the percentage of people who scored 700 or higher on the test. Find the percentage to the right of the 700 marker. ٣٦

1) Find the percentage of people who scored 700 or higher on the test. Find the z-score : z = 700-544 103 = 156 103 = 1.51.9345.0655 P(X>700)=P(Z>1.51)=1-P(Z<1.51)=1 -.9345 =.0655 2) Find the percentage of people who scored below 500 on the test. Find the percentage to the left of 500 2) Find the percentage of people who scored below 500 on the test. Find the z-score : z = 500-544 103 = -44 103 = - 0.43 0.3336 Answer : 0.3336 ٣٧

3) Find the percentage of people who scored between 500 and 800 on the test. Find the percentage between 500 and 800 3) Find the percentage of people who scored between 500 and 800 on the test. 500-544 Find the first z-score : z = = -44 103 103 = -0.43 Find the second z-score : z = 800-544 103 = 256 103 = 2.49 0.3336 0.9936 Area =.9936 -.3336 = 0.66 Example : The Soup Nazi charges, on the average, $4.50 for a cup of soup, and if you re lucky, some bread, with a standard deviation of $0.45. 4.50 What is the probability that our check will be more than $5.00? ٣٨

0.8665 0.1335 4.50 5.00 What is the probability that our check will be more than $5.00? P (X > 5 ) =P(Z >1.11)=0.1335 13.35 % Z = 5.00-4.50 0.45 = 1.11 Backward Normal Calculations We could find the observed value (x) of a given proportion in N(µ, σ) by unstandardizing the z- score. 1) State the problem 2) Draw a picture 3) Use the normal table to find the proportion closest to the one you need 4) Read off the z-value 5) Unstandardize x= µ+ zσ Example Find the value of z such that the probability of being less than z is 0.10. 1. z: P(Z < z) =.10.10 σ = 1 0 ٣٩

Example Find the value of z such that the probability of being less than z is.10. 1. z: P(Z < z) =.10 2..10 σ = 1 0 3. In the body of the normal table, find the closest value to.10. Once found, determine the z value. Closest is.1003 So z = -1.28 P(Z < -1.28) =.1003 Example Find the value of z such that the probability of being greater than z is.33. 1. z: P(Z > z) =.33 2. z: P(Z < z) = 1 -.33 =.67 σ = 1???.67.33 0 Example Find the value of z such that the probability of being greater than z is.33. 1. z: P(Z > z) =.33 2. z: P(Z < z) = 1 -.33 =.67.67.33 σ = 1 0 3. In the body of the normal table, find the closest value to.67. Once found, determine the z value. I found.6700 So z =.44 P(Z >.44) =.33 ٤٠

Example X = time Americans stir sugar into their iced tea X ~ N(12.3, 3.1) seconds (1) Find the percent of Americans who spend between 20 to 22 seconds in stirring sugar into their iced tea? i.e. P(20 < X < 22) Example X = time Americans stir sugar into their iced tea X ~ N(12.3, 3.1) Find P(20 < X < 22) = P(20-12.3 < Z < 22-12.3) 3.1 3.1 = P(2.48 < Z < 3.13) = P(Z < 3.13) - P(Z < 2.48) =.9991 -.9934 =.0057 Example X = time Americans stir sugar into their iced tea X ~ N(12.3, 3.1) (2) About 18.4% of Americans spend more than how many seconds stirring sugar into their iced tea? i.e. Find the value of X such that the probability of being greater than this value is.184. (1) z: P(Z > z) =.184 (2) z: P(Z < z) = 1 -.184 =.816 (3) From the normal table, z = 0.90 (4) So x = µ+zσ = 12.3 + 0.90(3.1) = 12.3 + 2.79 = 15.09 The person would have to stir 15.09 seconds. ٤١

Example X = IQ scores X ~ N(112, 9) Find the IQ score that replaces you in the top 2% of all scores. 1. z: P(Z > z) =.02 2. z: P(Z < z) = 1 -.02 =.98 3. From the normal table, z = 2.05 x = µ+zσ = 112 + 2.05 (9) = 130.45 Exercise The distribution of SAT Math scores is approximately normally distributed with mean 500 and standard deviation 100. 1. In what range do the middle 95% of all SAT Math scores lie? 2. What proportion of SAT Math scores are between 450 and 650? 3. If high school students having SAT Math scores in the top 10% of all scores are eligible for a certain scholarship, what is the lowest score a person eligible for the scholarship can have? ٤٢