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Descriptive Statistics Chapter 2 Descriptive Statistics

I. Section 2-1 A. Steps to Constructing Frequency Distributions 1. Determine number of classes (may be given to you) a. Should be between 5 and 15 classes. 2. Find the Range a. The Maximum minus the Minimum. 1) Use the TI-84 to sort the data. a) STAT – Edit – enter numbers into L1 b) STAT – SortA(L1) will put the numbers into ascending order. c) STAT – SortD(L1) will put the numbers into descending order. 3. Find the Class Width a. Range divided by the number of classes. 1) Always round UP!! a) Even if class width comes out to a whole number, go up one. 4. Find the Lower Limits a. Begin with the minimum value in your data set, and then add the class width to that to get the next Lower Limit. 1) Repeat as many times as needed to get the required number of classes.

5. Find the Upper Limits a. The Upper Limit of the first class is one less than the Lower Limit of the second class. 1) Add the class width to each Upper Limit until you have the necessary number of classes. 6. Find the Lower Boundaries a. Subtract one-half unit from each Lower Limit (Do NOT round these!) 7. Find the Upper Boundaries a. Add one-half unit to each Upper Limit. 8. Find the Midpoints of each class. a. The means of the Lower and Upper Limits (Do NOT round). 1) Could also use the means of the boundaries for this. 9. Frequency Distribution a. Place a tally mark in each class for every piece of data that fits there. b. Add up the tally marks – these are your frequencies for each class.

10. Relative Frequencies a. Divide the class frequencies by the total number of data points to find the percentage of the total represented by each class. 11. Cumulative Frequencies a. The total number of tallies for each class, plus all those that came before. 1) The cumulative frequency of the last class must equal the number of data points used.

EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes. 72 84 61 76 104 86 92 80 88 98 97 82 67 70 81 89 74 73 85 78 91 83 STAT – Edit – Enter these 30 numbers into L1 on your calculator STAT – SortA(L1) – 2nd 1 enters L1 into the parentheses. STAT – Edit – to see the new list, in order. 61 72 76 78 81 82 84 86 89 97 67 73 80 91 70 74 83 85 88 92 104

EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes. 61 72 76 78 81 82 84 86 89 97 67 73 80 91 70 74 83 85 88 92 104 Max Value: 104 Min Value: 61 Range: 104 - 61= 43 Class Width: 43/6 = 7.2 – Round UP to 8! LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. Minimum Value is the First Lower Limit 61 68 First Upper Limit is one less than 2nd Lower Limit Add Class Width Down 69 76 Add Class Width Down 77 84 85 92 Check to be sure that the Maximum Value fits in the last class. 93 100 101 108 Since 104 fits between 101 and 108, we are good. If the Maximum value does NOT fit into the last class, you did something wrong. DO IT AGAIN!! (You PROBABLY forgot to round UP on the class width)

EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes. 61 72 76 78 81 82 84 86 89 97 67 73 80 91 70 74 83 85 88 92 104 Max Value: 104 Min Value: 61 Range: 43 Class Width: 8 LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. Subtract one-half unit from lower limits to get lower boundaries 61 68 60.5 68.5 64.5 Find the mean of the limits (or boundaries) to find the midpoint of each class. 69 76 68.5 76.5 72.5 77 84 76.5 84.5 80.5 Add one-half unit to upper limits to get upper boundaries 85 92 84.5 92.5 88.5 93 100 92.5 100.5 96.5 101 108 100.5 108.5 104.5 CHECK – be sure that your class width is actually 8 units!!

EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes. 61 72 76 78 81 82 84 86 89 97 67 73 80 91 70 74 83 85 88 92 104 Max Value: 104 Min Value: 61 Range: 43 Class Width: 8 LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. Count how many data points fit in each class and enter that into the Frequency column 61 68 60.5 68.5 64.5 2 69 76 68.5 76.5 72.5 7 77 84 76.5 84.5 80.5 11 85 92 84.5 92.5 88.5 7 93 100 92.5 100.5 96.5 2 Block 7 – frequency of 3rd class 101 108 100.5 108.5 104.5 1

EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes. 61 72 76 78 81 82 84 86 89 97 67 73 80 91 70 74 83 85 88 92 104 Max Value: 104 Min Value: 61 Range: 43 Class Width: 8 LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. Relative Frequency is the class frequency divided by the total frequency. In this case, we have 30 pieces of data, so we divide by 30. 61 68 60.5 68.5 64.5 2 0.067 2/30 69 76 68.5 76.5 72.5 7 0.233 7/30 77 84 76.5 84.5 80.5 11 11/30 0.367 85 92 84.5 92.5 88.5 7 0.233 7/30 93 100 92.5 100.5 96.5 2 2/30 0.067 101 108 100.5 108.5 104.5 1 1/30 0.033 1.00 Relative Frequency column MUST add up to 1!! If it does NOT, you are wrong. DO IT AGAIN!!

EXAMPLE: Use the table of 30 numbers below to fill in a frequency distribution of 6 classes. 61 72 76 78 81 82 84 86 89 97 67 73 80 91 70 74 83 85 88 92 104 Max Value: 104 Min Value: 61 Range: 43 Class Width: 8 LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. Cumulative Frequency is the Frequency of each class, plus the classes that came before it. The last class must have a cumulative frequency that matches the number of data points 61 68 60.5 68.5 64.5 2 0.07 2 69 76 68.5 76.5 72.5 7 0.23 9 77 84 76.5 84.5 80.5 11 0.37 20 85 92 84.5 92.5 88.5 7 0.23 27 93 100 92.5 100.5 96.5 2 0.07 29 101 108 100.5 108.5 104.5 1 0.03 30

B. Steps to Constructing a Frequency Histogram 1. Label the horizontal axis with the class boundaries. 2. Label the vertical axis with the number of frequencies. 3. Draw a bar graph with bars that touch, using the frequencies from your frequency distribution.

C. Steps to Constructing a Relative Frequency Histogram 1. Label the horizontal axis with the class boundaries. 2. Label the vertical axis with the frequency percentages. 3. Draw a bar graph with bars that touch, using the relative frequencies from your frequency distribution.

Page 250, # 42 You are performing a study about the height of 20-29 year old men. A previous study found the height to be normally distributed, with a mean of 69.6 inches and a standard deviation of 3.0 inches. You randomly sample 30 men and find their heights (in inches) to be as follows: 72.1 71.2 67.9 67.3 69.5 68.6 68.8 69.4 73.5 67.1 69.2 75.7 71.1 69.6 70.7 66.9 71.4 62.9 64.9 68.2 65.2 69.7 72.2 67.5 66.6 66.5 64.2 65.4 70.0 Draw a frequency histogram to display these data points using seven classes. Find the mean and standard deviation of your sample. Compare the mean and standard deviation of your sample with those in the previous study. Discuss the differences.

Page 250, # 42 You are performing a study about the height of 20-29 year old men. A previous study found the height to be normally distributed, with a mean of 69.6 inches and a standard deviation of 3.0 inches. You randomly sample 30 men and find their heights (in inches) to be as follows: 72.1 71.2 67.9 67.3 69.5 68.6 68.8 69.4 73.5 67.1 69.2 75.7 71.1 69.6 70.7 66.9 71.4 62.9 64.9 68.2 65.2 69.7 72.2 67.5 66.6 66.5 64.2 65.4 70.0 Entering the 30 data points into the TI-84, using STAT and Edit, we can calculate the mean, standard deviation, and median. STAT, Calc, 1-Var Stats gives us what we need. The mean is 68.75, the standard deviation is 2.847, and the median is 69.

72.1 71.2 67.9 67.3 69.5 68.6 68.8 69.4 73.5 67.1 69.2 75.7 71.1 69.6 70.7 66.9 71.4 62.9 64.9 68.2 65.2 69.7 72.2 67.5 66.6 66.5 64.2 65.4 70.0 Max Value: 75.7 Min Value: 62.9 Range: 75.7 – 62.9 = 12.8 Class Width: 12.8/7 = 1.83 ≈ 1.9 Remember to ROUND UP!! We use one decimal place in our class width because we have one decimal place in our data points. LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. First Lower Limit is the Minimum Value!!! 62.9 64.7 First Upper Limit is one unit less than the 2nd Lower Limit (Remember, our units are tenths, not whole numbers). Add Class Width Down 64.8 66.6 66.7 68.5 Add Class Width Down 68.6 70.4 70.5 72.3 72.4 74.2 Make Sure Maximum Value Fits!! 74.3 76.1

72.1 71.2 67.9 67.3 69.5 68.6 68.8 69.4 73.5 67.1 69.2 75.7 71.1 69.6 70.7 66.9 71.4 62.9 64.9 68.2 65.2 69.7 72.2 67.5 66.6 66.5 64.2 65.4 70.0 LL UL LB UB MdPt Freq. Rel. Freq. Cum. Freq. Subtract one-half unit from lower limits to get lower boundaries. REMEMBER that our units are tenths!! One-half of a tenth is 5 hundredths (.05) 62.9 64.7 62.85 64.75 63.8 Find the mean of the limits (or boundaries) to find the midpoint of each class. 64.8 66.6 64.75 66.65 65.7 66.7 68.5 66.65 68.55 67.6 68.6 70.4 68.55 70.45 69.5 70.5 72.3 70.45 72.35 71.4 Add one-half unit to upper limits to get upper boundaries 72.4 74.2 72.35 74.25 73.3 74.3 76.1 74.25 76.15 75.2

72.1 71.2 67.9 67.3 69.5 68.6 68.8 69.4 73.5 67.1 69.2 75.7 71.1 69.6 70.7 66.9 71.4 62.9 64.9 68.2 65.2 69.7 72.2 67.5 66.6 66.5 64.2 65.4 70.0 Count how many data points fit in each class and enter that into the Frequency column. LL UL LB UB MdPt Freq. 62.9 64.8 62.85 64.75 63.8 2 64.9 66.8 64.75 66.65 65.7 5 66.9 68.8 66.65 68.55 67.6 8 68.9 70.8 68.55 70.45 69.5 8 Remember that sorting the data in the calculator can help with counting how many data values fall into each class. 70.9 72.8 70.45 72.35 71.4 5 72.9 74.8 72.35 74.25 73.3 1 74.9 76.8 74.25 76.15 75.2 1

Draw the histogram using the frequencies obtained from the table we just did. 8 5 2 1 62.85 64.85 66.85 68.85 70.85 72.85 74.85 76.85

The last part of the question was to compare the mean and standard deviation of your sample with those in the previous study. Discuss the differences. Our mean and standard deviation were 68.75 and 2.85. The previous study had a mean of 69.6 and a standard deviation of 3.0. This means that our sample of men was shorter than the previous study, but that they were also more closely bunched together in height.

D. Steps to Constructing an Ogive 1. Label the horizontal axis with the midpoints of each class. 2. Label the vertical axis with the total number of data points. 3. Place a dot at each midpoint that corresponds to that class’s cumulative frequency. a. This chart will always end at the total number of data points.

Assignments: Classwork: Pages 49-51 #1-8 All, 10-26 Evens Homework: Pages 52-53 #29-41 Odds