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Running head: EXCEL PROJECT

Excel Project: Public school expenditures and

It’s relationship to academic performance

Erin Brown

Seattle Pacific University

EDU 6976 Interpreting and Applying Educational Research II

Fall Quarter, 2008

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Data was collected from public school districts within all 50 of the United States, using

the 1997 Digest of Educational Statistics, an annual publication of the U.S. Department of

Education. This data was then analyzed using numerous statistical measures and the results

and interpretation of this analysis can be read as follows.

Part I Histograms, Box Plots, and Frequency Distributions

The data collected from the public schools of the 50 states includes: current

expenditures per pupil in average daily attendance, the pupil/teacher ratio, the estimated

annual salary of teachers, the percentage of all eligible students taking the SAT, the average

total score on SAT, the average math score on SAT, and the average verbal score on SAT.

We begin by looking at the distributions of the three continuous variables that represent

the treatment of the independent variable. That is, what the money was spent on

(expenditures, teacher ratio, and teacher salary). We can analyze these distributions by

putting the data into histograms and box plots.

Histograms show us the distribution of data. They allow us to view that data in a more

visual way. Let’s take a look at Figure 1 below. This histogram shows us the current

expenditures per pupil. You’ll note that the majority of students in the United States are listed

in the 4 to 6 bin. This means that $4,000 to $6,000 is allotted to the students to be spent on the

students.

<=0 (0, 2] (2, 4] (4, 6] (6, 8] (8, 10]

>100

5

10

15

20

25

30

35Frequency

Figure 1 – Histogram of Expenditures

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If we take a look at the box plot for this same expenditure variable (Figure 2), it also

gives us a visual display of the data, with the exception that this time it divides the data into

equal fourths or quartiles. Each quartile is equal to 25% and, therefore, all four quartiles are

equal to 100% of the data.

Lower Whisker

Lower Hinge Median

Upper Hinge

Upper Whisker

3.656 4.88175 5.7675 6.434 7.469

Figure 2 – Box Plot of Expenditures

Figure 2 shows us that the line to the left of the box is slightly longer than the line to the

right of the box. When box plot is not centered within the range, it shows skewness. We can

see that the distribution here is not quite normal. It skews slightly towards the positive or

upper end of the distribution. If we glance back at the histogram, we can see the reason for

this is the extremely high numbers in the fourth quartile.

Figures 3 and 4 give us a picture of the average teacher salary across America. The box

plot (Figure 4) shows that again the distribution is skewed. It’s positively skewed towards the

upper end of the distribution. The skewness is also more prominent this time.

<=10 (10, 20] (20, 30] (30, 40] (40, 50] (50, 60] (60, 70] >7005

101520253035 Frequency

Figure 3 - Histogram of Salary

0 2 4 6 8 10 12

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Figure 4 – Box Plot of Salary

The histogram (Figure 3) goes a long was to explaining why this skewness occurs.

Whereas, the majority of teachers in the United States are in the 30 to 40 thousand dollar

range, there is one state that throws the distribution off by offering on the average a much

higher pay.

Let us move on to the ratio data which is much more interesting. This information

refers to the average number of pupils (in the public elementary and secondary schools) per

teacher. When looking at the box plot (Figure 5), it looks positively skewed. It’s not that

impressive unless you take a look at the numbers.

Lower Whiske

rLower Hinge Median

Upper Hinge

Upper Whiske

r13.8 15.225 16.6 17.575 20.2

Figure 5 – Box Plot of Ratio

The lower whisker is 13.8 while the upper whisker is 20.2. This means that some states average

13 students per teacher and other states can have an average as high as 20 students per

teacher. Now, anyone who teaches in the classroom can tell you there is a huge difference in

6 8 10 12 14 16 18 20 22 24 26

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what can be accomplished with 13 students as opposed to 20 students. Also note that the

median is listed as 16 students.

<=12 (12, 15] (15, 18] (18, 21] (21, 24] (24, 27] (27, 30] >300

5

10

15

20

25

30Frequency

Figure 6 – Histogram of Ratio

If you take a look at the histogram (Figure 6), it gives you an obvious reason for this

skewness. Only 2 out of all 50 states are in the higher classroom ratio bins. These outliers, or

scores that are very high in comparison to others, have a direct effect on the mean. Because

these outliers are in the upper end of the data, they cause the distribution to be skewed

positively.

Now that we’ve looked at the data concerning the treatment of independent variable

and how the money was spent, let us now take a look at the outcome or dependent variables.

They are also continuous as they tell us about the eligible students who take the SAT, the

average verbal SAT score, the average math SAT score, and the total SAT score.

First, we’ll look at the data for the average total SAT scores in the United States. Both

the histogram (Figure7) and the box plot (Figure 8) show a slightly skewed distribution towards

the lower end of scores.

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<=775 (775, 845]

(845, 915]

(915, 985]

(985, 1055]

(1055, 1125]

(1125, 1195]

>119502468

101214161820

Frequency

Figure 7 – Histogram of Total Average SAT

Lower Whiske

rLower Hinge Median

Upper Hinge

Upper Whiske

r844 897.25 945.5 1032 1107

Figure 8 – Box Plot of Total Average SAT

This positive skew in the distribution could be due to the mode since there are 18 states that

average scores of 845 to 915. To give us more of an understanding, we need to see the

breakdown of data per each portion of the test.

400 600 800 1000 1200 1400 1600

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<=425 (425, 455]

(455, 485]

(485, 515]

(515, 545]

(545, 575]

(575, 605]

>60502468

101214161820

Frequency

Figure 9 - Histogram for Average Math SAT

<=375

(375, 400]

(400, 425]

(425, 450]

(450, 475]

(475, 500]

(500, 525]

(525, 550]

(550, 575]

(575, 600]

>6000

2

4

6

8

10

12

14

16Frequency

Figure 10 – Histogram for Average Verbal SAT

At first glance the histogram for the average math scores (Figure 9) seems to be the

reason, due to the positive skewness. The average verbal scores (Figure 10) may seem a little

inconsistent with low scores in the middle of the range, but overall show more strength than

the math averages. Looking deeper into the box plots of each, math SAT (figure 11) and verbal

SAT (figure 12); we see that verbal seems to be more of a problem.

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Figure 11 – Box Plot of Average Math SAT

Lower Whiske

rLower Hinge Median

Upper Hinge

Upper Whiske

r401 427.25 448 490.25 516

Figure 12 – Box Plot of Average Verbal SAT

Though the math box plot is positively skewed, the average scores on the verbal portion

are much lower than the math. The lowest math average score is 443 yet the lowest verbal

average score is 401. This is a good 40 point difference. The same could be said for the higher

scores. The highest math average is 76 points higher than the highest verbal average. Thus,

the lower overall average in the verbal scores has a greater effect on the total SAT average than

the skewness of the math scores.

Some of the most interesting data comes from the histogram (Figure 13) and the box

plot (Figure 14) of the percentage of eligible students that take the SAT. The box plot shows us

that only 0% to 20% of the students in 22 states who are eligible actually take the SAT. Now, is

200 300 400 500 600 700 800

Lower Whisker

Lower Hinge Median

Upper Hinge

Upper Whisker

443 474.75 497.5 539.5 592

200 300 400 500 600 700 800

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this due to the fact that only that many are eligible? Or is it due to the fact that most students

in these states do not plan on extending their education past high school?

<=0 (0, 20] (20, 40]

(40, 60]

(60, 80]

(80, 100]

>1000

5

10

15

20

25Frequency

Figure 13 – Histogram for Eligible Students Taking SAT

Lower Whiske

rLower Hinge Median

Upper Hinge

Upper Whiske

r4 9 28 63 81

Figure 14 – Box Plot for Eligible Students Taking SAT

I suppose that would lead to a completely different research study. What amazes me

about this information is that we can safely assume from this data that at the very least 80% of

the student population in these 22 states do not attend college. And what is even more

incredible when looking at the box plot, is that the lowest percentage in the range is 4% (the

lower whisker). Plainly speaking, 96% of the student population in that particular state has no

intention of officially continuing their education at least in the collegiate world.

-200 -150 -100 -50 0 50 100 150 200 250

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Figure 15 – Bar Graph of Regions

The last variable we need to consider is the categorical variable of the regions (Figure

15). Each of the 50 states is assigned to one of four regions: West (1), Midwest (2), South (3),

and the Northeast (4). Since 50 cannot be divided evenly by 4, it’s safe to predict that the

distributions will not be equal. This is important to keep in mind as this can have an effect on

the next portion of this research paper when we begin to take a closer look on how certain

regions differ in expenditures and the resulting test scores.

Part II Comparisons Using ANOVA

Breaking down our sample further into the four regions (West, Midwest, South, and

Northeast) may tell us more about their differences. To help us highlight these differences we

use the Analysis of Variance or ANOVA process.

ANOVA assesses whether the means of two or more groups are statistically different

from each other. Though there are other tests designed for this, ANOVA is the best choice for

our study due to its ability to analyze our four regions while controlling for Type I error.

When the expenditures per pupil are broken down into the four regions (Table 1.2), we

can see noticeable differences. The p-value of <.001 indicates that these differences are

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significant. When we conduct the Tukey test (Table 1.4), we can see that the differences are

significant between the Northeast and West and the Northeast and Midwest.

Comparing Expenditures by Region

Could this have something to do with the greater number of Northeastern states? After

all, there are more states listed under the Northeastern region than those of the West and

Midwest (Table 1.1). Though we should consider everything in this study, this doesn’t seem to

be the case in this particular situation. The South has fewer states listed within its region than

any of the others, yet no significant difference was determined when comparing it with the

Northeast.

The pupil/teacher ratio (Table 2.2) also indicates significant differences. Its p-value is

lower than the recommended .05. Thus, we again further the process by using the Tukey HSD

test (Table 2.4). This tells us that differences are especially significant between the Midwest

and West as well as the Northeast and West.

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Comparing Ratio by Region

If we isolate specific scores (Table 2.1) we can see the Midwest has one extreme score

or outlier that could be affecting the results. Also, and even more important, we can see two

overly inflated numbers among the West that could be the main reason for the differences

among the regions. So far, our analysis has shown more differences in the Western portion of

the country than anywhere else. Let’s continue on and see whether or not this trend continues.

Comparing Salary by Region

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The ANOVA table for the average teacher salary (Table 3.2) again indicates a significant

difference through its low p-value. The Tukey test (Table 3.4) pinpoints the exact location of

that difference. The Midwest and Northeast have significant differences among the average

teachers salary. Could this be due to the fact that the cost of living is lower (on average) within

the Midwest? Or could this be due to the fact that the cost of living is higher in the Northeaster

portion of the country? Also, if we take a look at the specific averages listed under these two

regions (Table 3.1), we can see one extreme number or outlier among the Northeastern region.

This inflated number may be the reason for the significant difference shown.

Though significant differences have been shown on all ANOVA tables thus far, several

adjustments could be made by individual states to level the playing field, so to speak. For

example, if California and Utah could make an effort to lower their pupil/teacher ratios to no

more than 20 students per instructor, there would be no significant differences among the four

regions. These two states have the power to equalize the pupil/teacher ratio across America.

We would then be better able to compare the equity in public school expenditures across

America by focusing on the other, more relevant, expenditures.

Now, let’s take a look at the percentage of all eligible students taking the SAT by region

(Table 4.2). Again, the p-value is lower than .05 indicating significant differences. The Tukey

test (Table 4.4) points out differences between the Northeast and both the West and Midwest.

Comparing Students Eligible to take SAT by Region

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Since the Northeast seems to be the common factor we look closer at those numbers

(Table 4.1). The Northeast numbers seem to be extreme in several directions. There are two

high percentages (80% and 81%) and two lower numbers (17% and 23%). Though not much can

be done to the high percentages except to make sure they are accurate, the two states with the

lowest percentages, Ohio and West Virginia, need to seriously focus on raising their

percentages.

Comparing Total Average SAT Scores by Region

Last, we look at the total average SAT scores by region (Table 5.2). The p-value is once

more low and points to significant differences. The Tukey test (Table 5.4) shows more

differences within this comparison then all the other previous tables. Differences among the

Midwest and all the other regions (West, South, and Northeast) were significant. Taking a

closer look at the Midwest (Table 5.1)which seems to be the common factor, we see that, with

the exception of one state, their numbers are consistently higher. Are students just smarter in

the Midwestern states?

What’s truly fascinating is when you look at the average total SAT score of the Midwest

region and see that it is 1048.08 (Table 5.3); and then compare it to the low percentage of

eligible students taking the SAT, a mere 12.58% (Table 4.3). Thus, fewer students are eligible to

take the test but those students who do take it are extremely bright (Charts 1 & 2). Why is this?

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West25%

Midwest27%

South25%

N-East23%

Chart 1 – Pie Chart of Total SAT Scores by Region

West22%

Midwest9%

South22%

N-East47%

Chart 2 – Pie Chart of Students Eligible to take SAT by Region

All these ANOVA tables have pointed to statistical significant differences within each

continuous variable in comparison of the four assigned regions. But what of the practical

significance?

In order to answer these questions we need to find the effect size. This is done by

calculating eta squared which is equal to the Sum of Squares Between Groups (SSb) divided by

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the Total Sum of Squares (SSt).

η2 = SSb / SSt

This is the simplest way of measuring the proportion of variance explained within an

ANOVA. If the effect size is small (.01), we will know that although the statistical difference may

be significant, the practical difference is rather small. Likewise, if the effect size is medium (.06)

or large (.14) this also explains the practical difference as opposed to the statistical (p-value)

difference (Table 6).

SSb SSt η 2 Practical diff.Expenditures 33.3015 91.0048 .42 largeRatio 107.355 251.682 .43 largeSalary 551.678 1729.63 .32 largeEligibility for SAT 17914.1 35095.1 .51 largeTotal SAT 129849 274308 .47 large

Table 6 – Effect Size

Table 6 shows a practical significant difference within all variables. It tells us that this

significance is extremely large.

There are a few assumptions to keep in mind when interpreting ANOVA data. First, we

use interval data. This is true in our study. Second, samples should be randomly sampled. I

think it’s safe to say our data came from a reputable source, the U.S. Department of Education.

Third, the samples must be independent of one another. In our study what effects one group

will not affect the others. Lastly, the Analysis of Variance must be of normal distribution. As

we’ve discussed early in Part I of this paper, several of our distributions are slightly skewed and

a few are most predominantly or significantly skewed. We need to keep this in mind when

dissecting ANOVA data.

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Part III Scatterplots and Linear Regression

Whereas correlation allows for “better than chance” predictions, regression takes this a

step further. Simple linear regression is a method for making such predictions. It can

sometimes point us to a possible cause.

To accomplish this we use scatterplots and regression equations. Scatter plots are used

to find confidence intervals. The slope and intercept of the regression line help us with the

regression equation.

Figure 16 – Scatterplot of expenditures and SAT scores

In Figure 16 we show the visual correlation between expenditures per pupil and the

average total SAT score. The scatterplot lets us see the relationship of these two variables in

the form of a line. This line passes through the scatterplot in a way that the average distance

from the line for each point in minimized.

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Table 7 – Expenditures and SAT scores

In Figure 16 we can see the elliptical cluster of corresponding points and the regression

line are at a negative angle but only slightly so. Table 7.1 lists the slope at negative 20.907 and

Table 7.2 tells us the intercept is 1089.41. These numbers are used to calculate the regression

equation.

Y = -20.907X + 1089.4

We can use this formula to find the possible SAT score given a certain amount of

expenditures. Unfortunately, there is no reason to make these calculations due to the

coefficient of determination (Table 7.3). Since this is 0.1448, this tells us that only 14%

variability in y can be explained by x. This seems to be an extremely weak relationship and not

worth our time to explore further.

When correlating the pupil/teacher ratio with the total SAT scores, we see zero

correlation as represented by the Pearson r in Table 8.3. This means that as one variable

changes, there is no related change to the other variable. In other words, as the pupil/teacher

ratio changes, the average SAT scores have no related change. It is important to note that

we’ve rounded our figures to 0%, even though there is a very slight change.

Table 8 – Ratio and SAT scores

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Again, the regression equation (Figure 17) is not much use to us. The correlation is just

too slight to be of any help.

Figure 17 – Scatterplot of ratio and SAT scores

This next set of data was the most interesting of all the simple linear regressions done in

this study. The average teacher’s salary and the current expenditures per pupil were charted

onto a scatterplot (Figure 18). We can clearly see the visual relationship in the form of a

positive elliptical slope. With the exception of one noticeable outlier, the points are clustered

closely on either side of the slope.

Figure 18 – Scatterplot of salary and expenditures

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Table 9.1 tells us the slope is 3.79. Table 9.2 points out a 12.44 intercept. But what

fascinates me the most is the coefficient of correlation (0.87) and the coefficient of

determination (.76) shown in Table 9.3. This correlation, according to Guilford’s interpretations

for values of r (Sprinthall, 2007), is strong and suggests a marked relationship. The coefficient

of correlation tells us that 87% of the variability in teacher’s salaries can be explained by

expenditures per pupil.

Table 9 – Expenditures and salary

This is where the regression equation (Figure 18) becomes extremely helpful. We can

take this equation and can use it to predict what a teacher’s salary is most likely to be based on

the amount of money spent per student (Table 9.4).

Table 9.4

Expenditures Equation Salary

X 3.792(X) + 12.436 Y

2.000 3.792(2) + 12.436 20.020

3.000 3.792(3) + 12.436 23.812

4.000 3.792(4) + 12.436 27.604

5.000 3.792(5) + 12.436 31.396

6.000 3.792(6) + 12.436 35.188

7.000 3.792(7) + 12.436 38.980

8.000 3.792(8) + 12.436 42.772

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9.000 3.792(9) + 12.436 46.564

10.000 3.792(10) + 12.436 50.356

According to our table, if a state were to spend on average two thousand dollars per

student, the average teacher salary in that state would be $20,020. Likewise, if a state were to

spend on average ten thousand dollars per student, the average teacher’s salary would be

$50,356. Though this doesn’t answer our original concern about whether or not school

spending and academic performance are statistically related, it would be helpful to any

teachers interested in relocating to another state or college students interested in teaching as a

possible career choice. But, of course, they need to keep in mind that though 87% of the

variability can be explained by the expenditures, there is another 13% that cannot be explained

by this.

Figure 19 – Scatterplot of ratio and salary

The last bivariate scatterplot between pupil/teacher ratio and average teacher salary

(Figure 19) is a great example of zero correlation. The regression line is parallel to the x axis.

With the exception of a few outliers, the points are scattered further apart and in a more

circular shape.

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Table 10 – Ratio and salary

The slope (Table 10.1) is -.003, the coefficient correlation (Table 10.3) is -0.00, and the

coefficient of determination is 0.00. Basically, this tells us that there is little to no relationship

between these two variables.

The only linear regression that showed a statistical significance in the relationships of

the variables was salary and expenditures. And, though this can be useful in many situations, I

don’t see how this can help us prove or disprove that school spending and academic

performance are statistically unrelated. I believe the fact that we didn’t find significant

correlations in the other scatterplots tells us much more.

Conclusions

None of the findings in this research study have done anything to alter the media’s claim

that school spending and academic performance are statistically unrelated. The histograms

brought to light several other concerns, such as low verbal SAT scores across the U.S and the

alarmingly low number of students taking the SAT in the Midwest. But these concerns cannot

be blamed on low expenditures. The ANOVA analysis showed statistically and practically

significant differences regionally, yet wouldn’t that be expected in a country so large and with

so many blended cultures? The linear regression data showed no correlations concerning

academic achievement (SAT scoring).

If this research was to be done a second time I would suggest that the 50 states be

broken into 5 regions evenly. This would cancel out any concerns regarding disproportion. I’d

also suggest a different form of testing. One standardized test to be used across the country

and to be mandatory for all high school seniors. This would give a true picture of how much is

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being learned. The SAT is only taken by those pupils wishing to attend college and that

excludes a large portion of the student body across America. If an alternate test cannot be

found, another possibility is to make the SAT mandatory to all high school seniors and wave the

fee charged for taking it the first time around.

Although several suggestions have been made regarding this cross-country research, I

would be more interested in this study being done exclusively statewide. The Washington state

WASL test could give us a wonderful standard per grade level and the state could be broken

down by regions. All students are already required to take this test so no extra testing would

be required. All the other information, ratio, salary, etc., is already on file due to the fact that it

had already been collected for the nationwide study. I believe focusing this research on a

smaller scale to one individual state would be much more effective and worthwhile.

Bibliography

National Center for Education Statistics (1997). Digest of Education Statistics. U.S. Department

of Education, 1997. Retrieved from http://nces01.ed.gov/pubs/digest97/index.html

Sprinthall, R.C. (2007). Basic Statistical Analysis (8th ed.). Boston, MA: Pearson Education, Inc.,

2007.

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