falenk.files.wordpress.com€¦ · web viewdata was collected for all 50 states on the average...

Excel Project 1

Running Head: EXCEL PROJECT

Excel Project: Public School Spending and its Relationship to SAT Scores

Kelly Falen

EDU 6976 Interpreting and Applying Educational Research II

Seattle Pacific University

Spring 2009

Excel Project 2

Part 1

Data was collected for all 50 states on the average expenditure per pupil in public school

and was compared to SAT scores for students in each state to determine whether there is a

relationship between spending and student achievement. In the data being compared and

analyzed is the current per pupil daily spending, student teacher ratio, estimated average annual

salary of the teachers, percentage of students eligible to take the SAT, and the average math,

verbal and total scores for those students who took the SAT. These data were extracted from the

1997 Digest of Education Statistics and will reflect information for the 1994-1995 school year.

Figures 1-3 are histograms, or a visual representation of data in terms of the frequency of

each data point, that depict the following average information: expenditure per student (in

thousands of dollars), student/teacher ratio and teacher salary. All three of these histograms show

that the data does not follow a normal distribution, but instead is positively skewed. In figure 1

the highest frequency of dollars spent per pupil is between five and six thousand. The histogram

in Figure 2 shows the highest frequency of student/teacher ratio is between 16 and 18 students

per teacher. Finally, Figure 3 indicates that the highest frequency of teacher salary is between 30

and 35 thousand dollars.

Excel Project 3

Figure I: Expenditure Per Pupil

Figure II: Student/Teacher Ratio

Excel Project 4

Figure III: Average Annual Teacher Salary

Next, Figures 4-7 indicate the percentage of students eligible to take the SAT, and the

average SAT scores per state, divided into math, verbal and total scores. These distributions do

not follow a normal curve and are bimodal, or there are two sets of scores which have high

frequency in these data sets. The percentage of eligible students in Figure 4 shows that the

highest frequency is between three and eighteen percent, with a smaller spike in frequency

between 63 and 78 percent. Figure 5 indicates that the highest frequency of math SAT scores fall

between 450 and 500 with another large spike between 525 and 550. The highest frequency of

verbal SAT scores, as shown in Figure 6, falls between 400 and 420 and again between 480 and

500, the numbers of scores that fall between these two ranges are the same. Finally, Figure 7

indicates that the highest frequency of total SAT scores fall between 850 and 900 and again

between 1000 and 1050.

Excel Project 5

Figure 4: Percentage of Students Eligible to take the SAT

Figure 5: SAT Scores – Math

Excel Project 6

Figure 6: SAT Scores – Verbal

Figure 7: SAT Scores - Total

Box plots are another way of presenting the data by showing a visual representation of

the distribution in quartiles, the middle 50th percent of the data, the median and any outliers that

may be skewing the data. The first box plot listed, Figure 8, has a median expenditure per pupil

of 5.7675 with a lower hinge of 4.88175 and an upper hinge of 6.434, indicating that the middle

50th percent falls between those two data points. This figure also shows that the data is

Excel Project 7

negatively skewed, as indicated by the larger amount of space on the left side of the box. There

is a trend in the first three box plots, Figures 9-10, that show outliers in the data sets indicating

abnormally high data points for the distribution of data given.

Figure 8: Expenditure per Pupil

In Figure 9 the median ratio of students per teacher is 16.6 with a lower hinge of 15.225

and an upper hinge of 17.575. There are also outliers in this data set and the majority of the

points fall below the median.

Figure 9: Student/Teacher Ratio

Figure 10 shows a median annual teacher salary of 33.2875 with a lower hinge of

30.9775 and an upper hinge of 38.5458. This data set indicates more data points above the

median as well as an outlier just beyond the end of the upper whisker.

Figure 10: Average Annual Teacher Salary (in thousands)

Excel Project 8

In Figure 11 the median is 28% of students who were eligible to take the SAT with the

majority of the data points above that percentage. Most of the data falls within the middle 50th

percent range of 9% to 63%.

Figure 11: Percentage of Students Eligible to take the SAT

Figures 12 – 14 detail the actual SAT Scores and are very similar in their distributions.

The median of the SAT math scores (Figure 12) is 497.5 with a lower hinge of 474.75 and an

upper hinge of 539.5. The median of the SAT verbal scores (Figure 13) is 448 with a lower hinge

of 427.25 and an upper hinge of 490.25. Finally the median for the total SAT scores 945.5 with a

lower hinge of 897.25 and an upper hinge of 1032. In all three of these box plots, the majority of

the scores fall above the mean.

Figure 12: SAT Scores – Math

Figure 13: SAT Scores – Verbal

Excel Project 9

Figure 14: SAT Scores – Total

Finally, Figure 15 shows the frequency distribution for the division of regions in the US

created for the purposes of this study in a simple bar chart format. All of the data indicated above

will be examined in further detail in the analysis section of this paper.

Figure 15: Frequency Distribution by Region

Part 2

The next step in analyzing the data provided is to conduct an analysis of variance

(ANOVA) to determine if there are statistical differences among regions. Once the determination

of statistical significance has been made through ANOVA, the next step is to determine which

regions are different from each other through Tukey’s HDS tests for each pair of regions and

variable. Then the means for each data set will indicate the direction of the difference. Once we

have established statistical significance, eta squared will indicate the effect size or practical

Excel Project 10

significance of each data set, giving a percentage of change for each variable that can be

explained by the differences in each region.

First, looking at expenditure per pupil across the four regions, ANOVA indicates that the

null hypothesis should be rejected. That is to say that there is a statistically significant difference

between the regions. The Tukey HSD test shows that the difference is between regions one and

four as well as regions two and four. According to the confidence intervals, region four is

outspending all three of the other regions. The mean expenditure per pupil for region four (in

thousands of dollars) is 7.21. Region one and two are spending an average of 5.54 and 5.74

respectively and region three is spending, on average, less per pupil than the other three regions

at 4.85. Eta squared is 0.42 which is considered to be a strong effect. This number indicates that

42% of the difference in expenditure per pupil can be explained by regional differences.

Expenditure Per Pupil

ANOVA Table 5%

Source SS df MS F Fcritical

p-value

Between

38.3015 3

12.767

11.143

2.8068

0.0000

Reject

Within52.703

446

1.1457

Total91.004

849

Estimates of Group MeansGroup Confidence Interval

15.5430

8 ±0.5976 95%

25.7415

8 ± 0.622 95%

3 4.847 ±0.6496 95%

47.2133

6 ±0.5758 95%

Tukey test for pairwise comparison of group means 1

r 4 2 2

Excel Project 11

n - r 46 3 3 q0 3.79 4 Sig Sig 4

T1.1710

9

Next, looking at the student/teacher ratio, ANOVA indicates that the null hypothesis

should be rejected because there is a statistically significant difference between the regions. The

Tukey HSD test shows that the significant difference is between regions one and two, regions

one and three and regions one and four. According to the confidence intervals, region one has a

higher ratio of students per teacher at 19.06, whereas regions two three and four have average

ratios of 16.2, 17.08 and 15.2 respectively. Region four has the lowest student/teacher ratio. Eta

squared is 0.43 which is considered to be a strong effect size. This number indicates that 43% of

the difference between student/teacher ratios can be explained by regional differences.

Student/Teacher Ratio

ANOVA Table 5%


p-value

Between

107.355 3

35.785

11.405

2.8068

0.0000

Reject

Within144.32

746

3.1375

Total251.68

249


119.061

5 ±0.9889 95%

2 16.2 ±1.0293 95%

317.081

8 ± 1.075 95%

4 15.2 ±0.9529 95%


r 4 2 Sig 2

Excel Project 12

n - r 46 3 Sig 3 q0 3.79 4 Sig 4

T1.9379

5

Next, looking at the average yearly salary for teachers in each region, again ANOVA

indicates that the null hypothesis must be rejected, that there is a significant difference between

the regions. The only regions that are statistically different in teacher salary according to Tukey’s

HSD test are regions two and four. Region four has the highest average annual teacher salary (in

thousands of dollars) at 39.60. Regions one, two and three have an average teacher salary of

34.71, 33.38 and 30.48 respectively. Region three has the lowest average annual teacher salary.

Eta squared is 0.32 which is considered to be a strong effect size. This number indicates that

32% of the differences between annual teacher salaries can be explained by regional differences.

Average Teacher Salary

ANOVA Table 5%


p-value

Between551.67

8 3 183.897.181

1 2.8068 0.0005 Reject

Within1177.9

5 46 25.608

Total1729.6

3 49Estimates of Group Means

Group Confidence Interval

134.712

5 ± 2.8251 95%2 33.381 ± 2.9405 95%

330.478

6 ± 3.0712 95%

439.596

1 ± 2.7223 95%

Tukey test for pairwise comparison of group means

Excel Project 13

1 r 4 2 2

n - r 46 3 3 q0 3.79 4 Sig 4

T5.5364

8

Next, looking at the percentage of students eligible to take the SAT test in each region,

ANOVA indicates that the null hypothesis must be rejected because there is a statistically

significant difference between the regions. Tukey’s HSD shows that the significant difference is

occurring between regions one and four as well as regions two and four. Region four has a much

higher average percentage of students who were eligible to take the SAT at 63.43. Regions one,

two and three had an average percentage of 30.38, 12.58, and 29.82 respectively. Region two has

the smallest percentage of students eligible to take the SAT. Eta squared is 0.51 which is

considered to be a strong effect size. This number indicates that 51% of the difference in

percentage of students eligible to take the SAT can be explained by regional differences.

Percent of Students Eligible to take the SAT

ANOVA Table 5%


p-value

Between

17914.1 3

5971.4

15.988

2.8068

0.0000

Reject

Within17181.

146 373.5

Total35095.

149

Excel Project 14


130.384

6 ±10.789 95%

212.583

3 ± 11.23 95%

329.818

2 ±11.729 95%

463.428

6 ±10.397 95%


r 4 2 2 n - r 46 3 3 q0 3.79 4 Sig Sig 4

T21.144

4

Next, looking at the SAT scores for math, ANOVA indicates that the null hypothesis

should be rejected because there is a statistically significant difference between the regions.

Tukey’s HSD shows that regions one and two, one and four, two and three as well as two and

four are significantly different from each other. The average SAT math score for region two was

the highest at 555.33. Region’s one, three and four had average scores of 508.54, 499 and 476.79

respectively. Region four had the lowest average SAT math scores. Eta squared is 0.52 which is

considered to be a strong effect size. This number indicates that 52% of the difference in math

SAT scores can be explained by regional differences.

SAT Scores -Math

ANOVA Table 5%


p-value

Between

41390.3 3

13797

16.783

2.8068

0.0000

Reject

Within37814.

346

822.05

Excel Project 15

Total79204.

649


1508.53

8 ±16.007 95%

2555.33

3 ± 16.66 95%

3 499 ±17.401 95%

4476.78

6 ±15.424 95%


r 4 2 Sig 2 n - r 46 3 Sig 3 q0 3.79 4 Sig Sig 4

T31.368

8

Next, looking at the verbal SAT scores for each region, ANOVA indicates that the null

hypothesis should be rejected because there is statistically significant difference between the

regions. Tukey’s HSD test shows the difference is between regions one and two, two and three as

well as two and four. Region two has the highest average verbal SAT score at 492.75. Region’s

one, three and four have an average verbal score of 455.31, 453.27 and 431.36 respectively.

Region four has the lowest average verbal SAT score. Eta squared is 0.41 which is considered to

be a strong effect size. This number indicates that 41% of the difference between the verbal SAT

scores can be explained by regional differences.

SAT Scores -Verbal

ANOVA Table 5%Source SS df MS F Fcritical p-

Excel Project 16

valueBetwee

n24731.

6 38243.

910.56

42.806

80.000

0Rejec

t

Within35898.

446 780.4

Total 6063049


1455.30

8 ±15.596 95%

2 492.75 ±16.233 95%

3453.27

3 ±16.954 95%

4431.35

7 ±15.029 95%


r 4 2 Sig 2 n - r 46 3 Sig 3 q0 3.79 4 Sig 4

T30.563

8

Finally, looking at total SAT scores for each region, ANOVA indicates that the null

hypothesis should be rejected because there is a statistically significant difference between the

regions. According to Tukey’s HSD test the difference is occurring between regions one and

two, two and three as well as two and four. Region two has the highest average total SAT score

at 1048.08. The average total scores for regions one, three and four are 963.85, 952.27 and

908.14 respectively. Region four has the lowest average total SAT score. Eta squared is 0.47

Excel Project 17

which is considered to be a strong effect size. This number indicates that 47% of the difference

in total SAT scores can be explained by regional differences.

SAT Scores -Total

ANOVA Table 5%


p-value

Between 129849 3 43283

13.783

2.8068

0.0000

Reject

Within 14445946

3140.4

Total 27430849


1963.84

6 ±31.285 95%

21048.0

8 ±32.563 95%

3952.27

3 ±34.011 95%

4908.14

3 ±30.147 95%


r 4 2 Sig 2 n - r 46 3 Sig 3 q0 3.79 4 Sig 4

T61.311

4

In comparing the data given between the regions some interesting conclusions can be

drawn. First, it is noteworthy that while region four has the highest expenditure per pupil as well

as the highest average annual teacher salary, that region also has the lowest SAT scores across

the board. Region two, which had the highest SAT scores, was neither highest nor lowest in

terms of spending per pupil or teacher salary. This indicates that the amount of money spent in

schools may not have a statistically significant relationship to student achievement on the SAT.

Excel Project 18

However, the region with the lowest expenditure per pupil and average annual teacher salary,

region 3, did not have the lowest SAT scores.

It is also interesting to note that region four had a markedly higher percentage of students

who were eligible to take the SAT, whereas region two had the lowest percentage of eligible

students. It is certainly worth further investigation to determine whether the number of students

eligible to take the SAT is impacting the data of this study in some way. For example, it would

be interesting to know what the criterion for eligibility is in each region. Perhaps only the most

successful students (based on a criterion different from SAT scores) were eligible to take the

SAT in region two, whereas region four may have had a more lax criterion, allowing far more

students to take the exam.

Across the data, the effect size was considered high for each area where statistical

significance was discovered. This indicates that for those areas where there is a statistically

significant difference, the practical significance is also high. The highest practical significance in

this study can be found in the comparisons between eligibility percentage in each region as well

as math SAT scores for each region – the percentage of difference in each of those areas which

can be explained by region is over 50%. The lowest area of practical significance in this study is

the difference between regions in terms of annual teacher salary with only 32% of the difference

being explained. The effect sizes of the remaining variables ranged between 41 and 47%.

Part 3

The final phase of analysis of this data is to determine whether there is a correlational

relationship between the key variables. In order to do this the correlational coefficient (r) and

coefficient of determination (r²) must be calculated. A visual representation of this data is in the

Excel Project 19

form of a scatter plot which will indicate at a glance whether correlations between the variables

exist as well as whether the correlation is positive or negative.

The first pair of variables being compared in this analysis is expenditure per pupil and

total SAT scores (Figure 16). The slope of the regression line for these variables is negative, -

20.892, indicating that there is a negative correlation between these two variables, which is to

say that as the expenditure per pupil increases, the total SAT score decreases. Taking a deeper

look at the data, the coefficient of correlation (r) is -0.380 indicating that there is a significant

correlation between these two variables because the critical value of r(48) is 0.361 for an alpha

level of .01. The coefficient of determination (r²) for these variables is 0.145 which indicates that

roughly 15% of the variable,variance in SAT scores, is influenced by expenditure per pupil. This

means that even though there is a correlation between the two variables, it is a low correlation.

Given the fact that only about 15% of the variance in the Y variable (SAT scores) is explained by

the X variable (expenditure per pupil) it is important to continue seeking other variables that may

have stronger correlations and can explain the remaining 85%.

Excel Project 20

Figure 16

The next pair of variables being compared in this analysis is student/teacher ratio and

total SAT scores (Figure 17). The slope of the regression line for these variables is positive,

2.683, indicating that there is a positive correlation between these two variables, which is to say

that as the student teacher ratio increases, the total SAT score also increases. Taking a deeper

look at the data, the coefficient of correlation (r) is 0.081 indicating that there is no significant

correlation between these two variables because the critical value of r(48) is 0.279 for an alpha

Excel Project 21

level of .05. The coefficient of determination (r²) for these variables is 0.006 which indicates that

roughly 0.6% of the variable, SAT scores, is influenced by student/teacher ratio. The fact that

there is no significant correlation between these two variables means that student/teacher ratio,

i.e. class size, is not a factor in determining success of students on the SAT test.

Figure 17

Excel Project 22

The next pair of variables being compared in this analysis is student/teacher ratio and

annual teacher salary (Figure 18). The slope of the regression line for these variables is negative,

-0.003, indicating that there is a negative correlation between these two variables, which is to say

that as the student teacher ratio increases, the annual teacher salary decreases slightly. Taking a

deeper look at the data, the coefficient of correlation (r) is -0.001 indicating that there is no

correlation between these two variables. The coefficient of determination (r²) for these variables

is 0.000 which indicates that almost 0% of the variable, annual teacher salary, is influenced by

student/teacher ratio. The fact that there is no significant correlation between these two variables

means that student/teacher ratio, i.e. class size, is not a factor in determining annual teacher

salary. It is interesting to note that there is not statistically significant correlation between teacher

salary and student/teacher ratio considering the fact that it would seem that school districts who

paid their teachers more would be unable to hire as many teachers as districts who pay less,

therefore increasing class size.

Excel Project 23

Figure 18

The next pair of variables being compared in this analysis is expenditure per pupil and

annual teacher salary (Figure 19). The slope of the regression line for these variables is

positive, .200, indicating that there is a positive correlation between these two variables, which is

to say that as the expenditure per pupil increases, the annual teacher salary also increases. Taking

a deeper look at the data, the coefficient of correlation (r) is 0.870 indicating that there is a

significant correlation between these two variables because the critical value of r(48) is 0.361 for

Excel Project 24

an alpha level of .01. The coefficient of determination (r²) for these variables is 0.757 which

indicates that roughly 76% of the variable, annual teacher salary, is influenced by expenditure

per pupil. This is considered a strong correlation given the fact that 76% of the Y variable

(annual teacher salary) is explained by the X variable (expenditure per pupil). This correlation is

not particularly surprising when it is considered that teacher salary is a component of expenditure

per pupil.

Figure 19

Excel Project 25

SUMMARY

Thorough analysis of the data collected indicates that there is not a statistically significant

relationship between school spending and SAT scores. The fact that the data for the variables

expenditure per pupil, student/teacher ratio and teacher class size is not normally distributed

(possibly due to outliers) and the data for the variables math, verbal and total SAT scores as well

as students eligible to take the SAT follows a bimodal distribution may be impacting the further

analyses.

The Tukey’s HSD tests showed significant differences between regions in all the

variables compared, however analysis of that data raised more questions than answers. While it

was clear that the regions with the highest spending did not have the highest test scores, the

regions with the lowest spending did not have the lowest scores either. This indicates that there

are other variables impacting test scores beyond spending. One possible variable indicated in this

analysis is eligibility to take the SAT test because it appears that the region (two) with the lowest

percentage of eligibility had the highest scores. This variable (eligibility) also had one of the

highest practical significance rates.

In an examination of correlation between key variables, it was determined that while

there is a significant correlation between school spending and SAT scores, that correlation only

accounts for approximately 15% of the variance, leaving 85% unexplained. This indicates again

that spending is not a major contributing factor in success on the SAT test.

Though school spending does not appear to have a statistically significant impact on

student success, based on the data in this study the conclusion can be reached that further

research needs to be done before we can actually determine what the true cause of student

success is for any practical purpose. In conclusion, it appears that that money may not be the

Excel Project 26

only variable that could impact student achievement on the SAT; therefore it would not be wise

to use this study as an argument to decrease school funding.

falenk.files.wordpress.com€¦ · web viewdata was collected for all 50 states on the average...

Documents