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Dr. Neal, WKU

MATH 183

Measurements

Statistics is the study of numerical data. Initially, we will be concerned with a specific measurement X taken from a population ! . For example, we may want to measure the number of credit hours each full-time student at WKU is taking this semester. After obtaining such measurements from either the entire population or from a sample of the population, we can begin to analyze the data in some way. The most elementary analysis involves computing the mean, median, mode, and standard deviation. Specifying the Population and the Measurement When studying a set of data, we first must know the population ! under consideration and the specific measurement X that we are analyzing. These terms should be properly defined before any study begins. If possible, we also should try to determine the actual population size N . Example 1. Last Spring, each student in a particular section MATH 132 were asked to give the number of credit hours they had earned so far in college. Below are the responses: Number of Credit Hours Earned 16 46 18 17 48 18 84 50 16 18 16 89 78 76 78 124 92 76 Sort the values in increasing order, in a frequency chart, and according to undergraduate classification. Solution. Although not explicitly asked for in the problem, we first should state the population ! , its size N , and the measurement X before analyzing the data. Here,

! = Students in this specific MATH 132 in Spring 2007, (A specific person in ! is denoted by ! , the small Greek omega.) From the data, we can determine that the population size is N = 18. Then, X = total college credit hours earned (before Spring 2007). (If person ! has earned 48 hours, then we can write X(! ) = 48 .) We now sort the data into increasing (or decreasing) order: 16 16 16

17 18 18

18 46 48

50 76 76

78 78 84

89 92 124

Next, we can make a frequency chart that displays all the possible measurements (number of hours earned) and the numbers of students (frequencies) having each possible measurement.

Dr. Neal, WKU

Frequency Chart Hours Earned (Measurement X) 16 17 18 46 48 50 76 78 84 89 92 124

# of Students (Frequencies) 3 1 3 1 1 1 2 2 1 1 1 1 N = 18

We also can group the data into categories that only display the numbers (or percentages) of students having certain ranges of measurements. But in this case, we lose each specific measurement. Range of Hours Fr: [0, 30) So: [30, 60) Jr: [60, 90) Sr: [90, !)

Number of Students 7 3 6 2

Percentage 38.89% 16.67% 33.33% 11.11%

Using the TI-83/84 Calculator Enter the raw data from Example 1 into your calculator, then use the calculator to (i) sort the data into increasing order; (ii) make a histogram with range [0, 120] with bins of length 30; (iii) compute the basic statistics. (iv) Do the same by entering the frequency chart data. Solution. We first must enter the data into a list. Press STAT, then press 1 to bring up then STAT Edit screen. In order to clear any data that might be in the lists, press the Up arrow to highlight L1, press CLEAR, then press ENTER. If necessary, highlight other lists, press CLEAR, and press ENTER. Then move the cursor back under list L1.

STAT EDIT 1

List Editor

Highlight L1, press CLEAR, then press ENTER

Enter data into an empty list.

Dr. Neal, WKU

To enter the data into list L1, type 16, press ENTER; type 12, press ENTER; continue until all the hours earned are entered into L1. Then press 2nd QUIT to return to the Home screen. From the STAT EDIT screen, retrieve the command SortA(, then enter the command SortA(L1. Then view the sorted data in the STAT Edit screen.

Enter raw data. Then press 2nd QUIT.

STAT EDIT 2

Enter SortA(L1

View sorted data.

(L1 is 2nd 1) Making a Histogram

To make a histogram of the data, we must adjust the WINDOW and STAT PLOT settings. Here, the range of measurements is 0 (Xmin) to 120 (Xmax) with bins of length 30 (Xscl). Always make Ymin = 0, but adjust Ymax as necessary to see each bar. Adjust the STAT PLOT settings to a histogram (3rd type), set the Xlist to L1 with frequencies 1, and press GRAPH. Press TRACE to see how many measurements are in each bin.

Adjust WINDOW

STAT PLOT 1

Adjust settings to 3rd type and plot L1 with Freq 1

Press GRAPH and then TRACE.

7 are in [0, 30)

3 are in [30, 60)

6 are in [60, 90)

2 are in [90, 120)

Basic Statistics

STAT CALC 1

Enter 1–VarStats L1

Output

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For this class, the average number of credit hours earned is 960/32 " 53.33. The median number of hours earned was 49. (All the statistics will be defined and given proper notation later.)

Dr. Neal, WKU

Using the Data in the Frequency Chart Now we will enter the distinct measurements into list L2 and enter the frequencies into list L3. Adjust the WINDOW as before, then adjust the lists in the STAT PLOT settings to graph L2 versus L3.

Enter frequency chart data into L2 and L3.

STAT PLOT 1

Adjust settings to 3rd type and plot L2 with FREQ L3

Press GRAPH.

To compute the statistics, enter the command 1–VarStats L2,L3 where L2 represents the measurements and L3 represents the frequencies.

STAT CALC 1

1–VarStats L2, L3

Output

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Example 2. How tall are the basketball players at UConn? Data is given below. Enter the frequency charts into your calculator and compute the basic statistics for the women, men, and all. Heights 63 65 66 67 68 70 71 72 73 74 75 76 77 79 80 84

# Players 1 1 1 2 1 2 2 1 3 2 2 2 2 4 2 1 N = 29

# Women 1 1 1 2 1 2 1 0 3 0 1 1 1 0 0 0 N1 = 15

# Men 0 0 0 0 0 0 1 1 0 2 1 1 1 4 2 1 N2 = 14

Dr. Neal, WKU

Solution. Let ! = All UConn basketball players and let X = height in inches. In all, there are N = 29 players. But ! breaks up into two sub-populations: !1 = women basketball players with N1 = 15 players; and ! 2 = men basketball players with N2 = 14 players. Enter the measurements into L1 and the successive frequencies into lists L2, L3, and L4. Then enter the command 1–Vars Stats L1,L2 to compute the overall stats, and enter 1–Vars Stats L1,L3 and 1–Vars Stats L1,L4 to compute the stats for the women and the men individually.

All in L1,L2

1–Vars Stats L1,L2

Overall Stats

Women in L1,L3

1–Vars Stats L1,L3

Women Stats

Men in L1,L4

1–Vars Stats L1,L4

Men Stats

The women average 70.27 inches (roughly 5' 10 1 4 "), the men average 77.07 inches (roughly 6' 5"), and the overall average height is 73.55 inches (roughly 6 ' 1 12 ")....