Frequency Distributions, Central Tendency and Variability

Frequency Tables, Central Tendency and Variability

1. Remember the SPSS file you created with New Mexico State Highway and Transportation Department’s data for 2003 traffic deaths by county in the tutorial? Open that file (it should be named nmdeath.sav) and use SPSS to construct a simple frequency table of the “deaths” variable.

County # of Deaths County # of Deaths County # of Deaths Bernalillo 54.00 Roosevelt 2.00 Catron 3.00 Harding 0.00 Sandoval 16.00 Chaves 4.00 Hidalgo 3.00 San Juan 37.00 Cibola 12.00 Lea 11.00 San Lincoln 3.00 Colfax 10.00 Miguel 8.00 Los Curry 6.00 Santa Fe 16.00 Alamos 1.00 De 3.00 Sierra 6.00 Luna 10.00 Doña 26.00 Socorro 11.00 McKinley 34.00 Eddy 14.00 Taos 5.00 Mora 4.00 Grant 5.00 Torrance 8.00 Otero 13.00 Guadalupe 4.00 Union 4.00 Quay 7.00 Valencia 14.00 Rio Arriba 13.00 a) Copy the frequency table from your SPSS output by right-clicking on the table and clicking on “copy object”. Paste the table in the space below (the table will likely drop to the next page)

Deaths

Cumulative Frequency Percent Valid Percent Percent Valid .00 1 3.0 3.0 3.0 1.00 1 3.0 3.0 6.1 2.00 1 3.0 3.0 9.1 3.00 4 12.1 12.1 21.2 4.00 4 12.1 12.1 33.3 5.00 2 6.1 6.1 39.4 6.00 2 6.1 6.1 45.5 7.00 1 3.0 3.0 48.5 8.00 2 6.1 6.1 54.5 10.00 2 6.1 6.1 60.6 11.00 2 6.1 6.1 66.7 12.00 1 3.0 3.0 69.7 13.00 2 6.1 6.1 75.8 14.00 2 6.1 6.1 81.8 16.00 2 6.1 6.1 87.9 26.00 1 3.0 3.0 90.9 34.00 1 3.0 3.0 93.9 37.00 1 3.0 3.0 97.0 54.00 1 3.0 3.0 100.0 Total 33 100.0 100.0

1 Marley EdPsych 511, HW#1 Answer the following questions: a) How many counties had 4 deaths? 4 b) What percentage of counties had 13 deaths? 6.1% c) What percentage of counties had 16 or fewer deaths? 87.9%

2. With the data above, and SPSS, build a grouped frequency table using the following intervals: 0 to 4, 5 to 9, 10 to 14, 15 to 19, 20 to 24, 25 to 29, 30 to 34, 35 to 39, 40 to 44, 45 to 49, 50+. If you’re not sure how to do this check the tutorial. When you have completed your table copy it from the SPSS output by right-clicking on the table and clicking on “copy object”. Then paste the table in the space below.

Deaths (Banded)

Cumulative Frequency Percent Valid Percent Percent Valid <= 4.00 11 33.3 33.3 33.3 5.00 - 9.00 7 21.2 21.2 54.5 10.00 - 14.00 9 27.3 27.3 81.8 15.00 - 19.00 2 6.1 6.1 87.9 25.00 - 29.00 1 3.0 3.0 90.9 30.00 - 34.00 1 3.0 3.0 93.9 35.00 - 39.00 1 3.0 3.0 97.0 50.00+ 1 3.0 3.0 100.0 Total 33 100.0 100.0

Based on the completed frequency table above, Answer the following questions: a) What percentage of counties had between 15 and 19 fatalities? 6.1% b) What percentage of counties had 9 or fewer fatalities? 54.5% c) If the State determined that any county with 15 or more fatalities had to implement a Fatality Reduction Program, how many counties would have to implement the program?6 d) What percentage of the counties would that be? 18.1%

2 Marley EdPsych 511, HW#1 3. In a recent survey of Albuquerque Public School Teachers, they were asked to report how many years they had been teaching in their current school. Below are responses from 10 teachers. Using these data, perform the calculations necessary to complete the table and answer the following questions.

Teacher Years at Current X-µ (X-µ)2 Number School (X) 1 6 .40 .16 2 10 4.40 19.36 3 2 -3.60 12.96 4 8 2.40 5.76 5 4 -1.60 2.56 6 6 .40 .16 7 5 -.60 .36 8 7 1.40 1.96 9 2 -3.60 12.96 10 6 .40 .16 n= ∑=56 ∑=0 ∑=SS=56.4 a) What is the mean of the distribution? 5.6 b) What is the mode of the distribution? 6 c) What is the median of the distribution? 6 d) What is the sum of squares? 56.4 e) Calculate the variance for the sample. 56.4/9 = 6.26 f) Calculate the sample standard deviation. 2.50 g) Using the computational formula that was provided in class, calculate the variance. h) Check answers 3a – 3c and 3e-3f using SPSS.

Descriptive Statistics

N Minimum Maximum Mean Std. Deviation Variance VAR00001 10 2.00 10.00 5.6000 2.50333 6.267 Valid N (listwise) 10

4. The following exercise is to explore the effects of a constant on measures of central tendency and variability. The data set 'PS1 Essays.sav' contains scores for 228 Political Science students on an essay exam. The data set contains one variable called “raw”, which is the students' raw scores, that is, out of 50 points. Create a new variable called “add2” by going to the transform menu, select compute, typing add2 in the target variable box, and then in the right hand box add two to the raw score (i.e., raw + 2). Create a third variable called “times two” by repeating the process only this time name the target variable times2 and multiple the raw variable by two.

3 Marley EdPsych 511, HW#1 Complete the following SPSS activity and answer the questions below. Be sure to provide a copy of your output. a) Download the "Essays" data set from the course website. b) Using SPSS, calculate descriptive statistics for the three variables (raw, add2, times2).

Descriptive Statistics

N Minimum Maximum Mean Std. Deviation Raw Scores 228 25.00 50.00 40.4868 4.97481 add2 228 27.00 52.00 42.4868 4.97481 times2 228 50.00 100.00 80.9737 9.94962 Valid N (listwise) 228 c) Create a histogram of each variable. Using the same technique you used with the frequency tables copy and paste the three histograms in the space below.

c 30 n e u q e r F 20

Mean = 40.4868 Std. Dev. = 4.97481 N = 228 0 25.00 30.00 35.00 40.00 45.00 50.00 Raw Scores

40 y c

n 30 e u q e r F 20

Mean = 42.4868 Std. Dev. = 4.97481 N = 228 0 25.00 30.00 35.00 40.00 45.00 50.00 55.00 add2

4 Marley EdPsych 511, HW#1 50

c 30 n e u q e r F 20

Mean = 80.9737 Std. Dev. = 9.94962 N = 228 0 50.00 60.00 70.00 80.00 90.00 100.00 times2 d) Save your output to your own floppy disk.

4e) Using the information above, answer these questions. i) What are the means of "raw", "add2", and "times2"? 40.4868, 42.4868, & 80.9737

ii) What are the Standard Deviations of "raw", "add2", and "times2"? 4.97481, 4.97481, 9.9496

4f) Given your results, complete the following:

i) Summarize the effects of addition and multiplication of a constant on the mean of a sample.

Addition shifts the mean of the distribution on the number line by two places. Multiplication shifts the mean of the distribution by the number multiplied, in this case two.

ii) Summarize the effects of addition and multiplication of a constant on the standard deviation of a sample.

Addition had no effect on the standard deviation. Multiplication doubled the standard deviation.

iii) Speculate about the effect on the variance. Addition had no effect on the variance. Multiplication by two increased the variance by a factor of four.

5 Marley EdPsych 511, HW#1 iv) Discuss the effects of each transformation in terms of the distribution itself. (i.e. its shape and placement on the number line). Be sure to look at your histograms for this item.

Addition did not change the variability. Multiplication increased the variability.

6 Marley EdPsych 511, HW#1