In response to recent controversy over equity in public school expenditures, the Digest of
Education Statistics conducted a study focusing on the relationship between school expenditure
and academic performance. Data is included from all 50 states with results and interpretations
of this data shown below in the form of several statistical measures.
Part I
Figure 1 shows the 2005-2006 school year expenditure per pupil in average daily attendance
for both public elementary and secondary schools. The data shows a positive skew as the
distribution is skewed to the right on the histogram. The highest distribution falls within the
$8,000 to $10,000 range.
Figure 1: Histogram of the Current Expenditure Per Pupil in Average Daily Attendance in Public Elementary and Secondary School
Figure 2 Histogram shows the distribution of students in elementary and secondary schools
who are eligible for free or reduced priced lunch. This distribution shows a skewed distribution
to the right demonstrating a positive skew between the values of 30 and 40. The highest
frequency, or mode, is between the values of 30 and 35.
Figure 2: Histogram of the percentage of students eligible for free and reduced lunch 2006--07
Figure 3 histogram displays data for the estimated average annual salary of public
teachers for both elementary and secondary schools. The data is skewed slightly to the right in
a positive direction. The highest frequency for salary is between $40,000 to $45,000.
Figure 3: Histogram of the Estimated Average Annual Salary of Teachers in Public Elementary and Secondary Schools in 2005-2006
Figure 4 shows the histogram displaying data for the average SAT verbal (reading) scores
from the 2005-2006 school year. This distribution of scores is positively skewed to the right.
The SAT verbal (reading) score with the highest frequency is found in the score range of 490 to
500.
Figure 4: Histogram of the Average Score per State on the Verbal (Reading) SAT 2005-06
As an addition to histograms, box plots are created to view the data. Box plots are divided
into quartiles and allow each variable to be displayed. We can display the variables in such a
way as to identify any outliers that would skew the data. We can also clearly see the median
Heading 1 for Box Plots
Expenditures
The box plot for figure 1 displays the data for the current expenditure per pupil in average
daily attendance in public elementary and secondary schools for the 2005-06 school year, and
shows that 50% of the expenditure falls between $8,639.00 and $11,426.00. The median
expenditure per student is $9,805.00. The range of lowest to highest expenditure is from
$5,960.00 to $14,277.00 per student. The outliers are shown at $15,759.00, $16,511.00, and
$18,339.00. There is a positive skew as the median skews toward the lower hinge.
Lower Whisker
Lower Hinge Median
Upper Hinge
Upper Whisker
35607 42179.5 45575 53276.5 61372
Figure 1:Box Plot for the Current Expenditure Per Pupil in Average Daily Attendance in Public Elementary and Secondary Schools, 2005-06
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Heading 2 for Box Plots
Percentage of Students Eligible for Free or Reduced-Priced Lunch
Figure 2 shows the percentage of students eligible for free or reduced-price lunch from
2006-2007. The data shows that 50% fall between 32.1% and 47.8%. The median percentage is
37.5% with the minimum value at 17.7% and the maximum value at 67.5%. The median is not
the same distance from the lower and upper hinge and it exhibits a positive skew to the data.
There is an outlier present at 0%, which is due non-reported percentage by that state.
Lower Whisk
er
Lower
Hinge
Median
Upper
Hinge
Upper Whisk
er17.7 32.1 37.5 47.8 67.5
-40 -20 0 20 40 60 80 100 120
Percent of students in elementary and secondary schools who are eligible for free or reduced-price lunch, by state or jurisdiction: School year 2006-07
Percentage of students eligible for free or reduced-price lunch
Figure 2: Percentage of Students Eligible for Free or Reduced-Priced Lunch 2006-07
Heading 3 Box Plot
Annual Average Teacher Salary
Figure 3 shows that 50% of the salaries of the annual salary for public elementary and
secondary teachers fall between $42,179.50 and $53,276.50. The minimum value falls at
$35,607.00 and the maximum value for average salary falls at $61,372.00. The median average
salary is $45,575.00. The median is not the same distance from the lower and upper hinge and
it exhibits a positive skew to the data. When looking at the box plot there does not appear to be
any outliers.
Lower Whiske
rLower Hinge
Median
Upper Hinge
Upper Whiske
r
3560742179.
5 4557553276.
5 61372
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Average Annual Salary for Public Elementary and Secondary Teachers 2005-2006
Average Annual Salary in US Dollars
Figure 3: Box Plot for the Average Annual Salary of Public Elementary and Secondary Teachers 2005-2006.
Heading 4 for Box Plots
SAT Verbal
Figure 4, we can see that 50% of the average verbal SAT scores fall between 498 and 569. The
median score for the verbal SAT is 523. The minimum value falls at 482 and the maximum
value for the verbal SAT is 610. When looking at this data, it doesn’t appear to be any outliers.
The median is shifted towards the lower hinge indicating a positive skew in this data.
Lower Whiske
rLower Hinge Median
Upper Hinge
Upper Whiske
r482 498 523 569 610
Figure 4. Box Plot for the Average Verbal SAT Score from 2005-06
Part Two
In part two of this study, we looked at differences between the four regions: West, Midwest,
South, and Northwest. This differences in regions are for our four areas: Current Expenditure
per Student in Average Daily Attendance, Percentage of Student Free and Reduced-Priced
200 300 400 500 600 700 800 900
Lunch, Estimated Average Teacher Salary at Elementary and Secondary Levels, and Verbal
SAT Score.
The ANOVA is used to determine differences between the groups. Each ANOVA is run for
each variable with the alpha level of .05. Also the Tukey test is used to determine if there are
significant differences between the group means
Expenditure
The ANOVA using an alpha level of .05 for current expenditure per pupil in the daily
attendance in public elementary and secondary schools, 2005-06 (Table 1) shows several
different findings. The F ratio is significant at 9.75, and the p-value is less than .05 showing
unequal variances among the groups. The null must be rejected showing there are no
differences between the groupings because the F ratio of 9.75 was greater than the critical value
of F which is 2.802.
Also, the ANOVA effect size is found by taking the SSb at 120097648.8 divided by the SSt at
313050750.6 which calculates to the effect size of .383 (eta squared). This is statistically and
practically significant.
Finally, the Tukey test for pair wise comparison for the group means shows a significant
difference between the Northeast, and the groups from the West and Midwest.
Table 1: Current Expenditure Per Pupil in Average Daily Attendance in Public Elementary and Secondary Schools, 2005-06
ANOVA Table 5%
Source SS df MS FF
critical p-value
Between1.2E+
08 3 4E+07 9.75122.80
24 0.0000Reje
ct
Within1.9E+
08 47 4E+06
Total3.1E+
08 50
Estimates of Group MeansGroup Confidence Interval
Northwest
9244.92 ± 1130.5 95%
Midwest9905.
42 ± 1176.7 95%
South9720.
88 ± 988.61 95%Northea
st13601
.4 ± 1358.7 95%
Tukey test for pairwise comparison of group means
Northwe
st
r 4Midwes
t Midwe
st
n - r 47 South Sout
h
q0 3.79Northea
st Sig Sig Northea
st
T2559.
73
SES
The ANOVA using an alpha level of .05 for Percentage of Students Eligible for Free or
Reduced-Priced Lunch 2006-07 (Table 2) shows several different findings. The F ratio is
significant at 14.87, and the p-value is less than .05 showing unequal variances among the
groups. The null must be rejected showing there are no differences between the groupings
because the F ratio of 14.87 was greater than the critical value of the critical F which is 2.8068.
Also, the ANOVA effect size is found by taking the SSb at 2732.83 divided by the SSt 5550.3
which calculates to the effect size of .492 (eta squared). This is statistically and practically
significant.
Finally, the Tukey test for pair wise comparison for the group means shows a significant
difference between the Midwest, and the South, as well as between the South and the
Northeast.
Table 2: Percentage of Students Eligible for Free or Reduced-Priced Lunch 2006-07
ANOVA Table 5%
Source SS df MS FF
critical p-value
Between2732.
83 3 910.94 14.8712.80
68 0.0000Reje
ct
Within2817.
7 46 61.254
Total5550.
53 49
Estimates of Group MeansGroup Confidence Interval
Northwest
39.5667 ± 4.5478 95%
Midwest34.42
5 ± 4.5478 95%South 49.13 ± 3.8209 95%
53Northea
st29.77
78 ± 5.2513 95%
Tukey test for pairwise comparison of group means
Northwe
st
r 4Midwes
t Midwe
st
n - r 46 South SigSout
h
q0 3.79Northea
st SigNorthea
st
T9.887
5
Salary
The ANOVA using an alpha level of .05 for the estimated annual salary of teachers in public
elementary and secondary schools in 2005-06. (Table 3) shows several different findings. The
F ratio is significant at 3.45, and the p-value is .0238 showing unequal variances among the
groups. The null must be rejected showing there are no differences between the groupings
because the F ratio of 3.45 was greater than the critical value of F which is 2.802.
Also, the ANOVA effect size is found by taking the SSb at 434910474.8 divided by the SSt at
240957680 which calculates to the effect size of .180 (eta squared). This is statistically and
practically significant.
Finally, the Tukey test for pair wise comparison for the group means shows there is not a
significant difference between the four regions for the estimated annual salary of teachers in
public elementary and secondary schools in 2005-06.
Table 3: Estimated Annual Salary of Teachers in Public Elementary and Secondary Schools in 2005-06
ANOVA Table 5%Source SS df MS F Fcritical p-value
Between4.3E+
08 31E+0
8 3.45052.802
4 0.0238Reje
ct
Within 2E+09 474E+0
7
Total2.4E+
09 50
Estimates of Group MeansGroup Confidence Interval
West47223.
4 ±3616.6 95%
Midwest46312.
5 ±3764.3 95%
South45717.
4 ±3162.6 95%
Northeast
53864.9 ±
4346.6 95%
Tukey test for pairwise comparison of group means West
r 4 Midwest Midwe
st
n - r 47 South Sout
h
q0 3.79Northea
st Northea
st
T8188.7
2
Verbal
The ANOVA using an alpha level of .05 for the Average Verbal SAT Score from 2005-06.
(Table 4) shows several different findings. The F ratio is significant at 12.00, and the p-value is
less than .05 showing unequal variances among the groups. The null must be rejected showing
there are no differences between the groupings because the F ratio of 12.00 was greater than the
critical value of F which is 2.802.
Also, the ANOVA effect size is found by taking the SSb at 30986 divided by the SSt 40450.8 at
which calculates to the effect size of .766 (eta squared). This is statistically and practically
significant.
Finally, the Tukey test for pair wise comparison for the group means shows there is not a
significant difference between the four regions for the estimated annual salary of teachers in
public elementary and secondary schools in 2005-06.
ANOVA Table 5%Source SS df MS F Fcritical p-value
Between 30986 3 10329 12.0012.802
4 0.0000Reje
ct
Within40450.
8 47 860.66
Total71436.
8 50
Estimates of Group MeansGroup Confidence Interval
Northwest
528.692 ± 16.369 95%
Midwest 576.5 ± 17.037 95%
South526.76
5 ± 14.314 95%Northeas
t 504 ± 19.673 95%
Tukey test for pairwise comparison of group means
Northwe
st
r 4 Midwest SigMidwe
st
n - r 47 South SigSout
h
q0 3.79Northea
st Sig Northea
st
T37.062
3
SES
ANOVA Table 5%
Source SS df MS FF
critical p-value
Between2732.
83 3 910.94 14.8712.80
68 0.0000Reje
ct
Within2817.
7 46 61.254
Total5550.
53 49
Estimates of Group MeansGroup Confidence Interval
Northwest
39.5667 ± 4.5478 95%
Midwest34.42
5 ± 4.5478 95%
South49.13
53 ± 3.8209 95%Northea
st29.77
78 ± 5.2513 95%
Tukey test for pairwise comparison of group means
Northwe
st
r 4Midwes
t Midwe
st
n - r 46 South SigSout
h
q0 3.79Northea
st SigNorthea
st
T9.887
5
Confidence Interval for Slope (1-) C.I. for 1
95%-
0.00259 + or - 0.00138
Confidence Interval for Intercept (1-) C.I. for 0
95% 658.198 + or - 66.2714
r20.225
4 Coefficient of Determination
r
-0.474
8 Coefficient of Correlation
s(b1)0.000
68 Standard Error of Slope
s(b0)32.97
78 Standard Error of Intercept
Scatter Plot, Regression Line and Regression Equation
30,000.0035,000.0040,000.0045,000.0050,000.0055,000.0060,000.0065,000.000.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
f(x) = − 0.00258512605430003 x + 658.197604368544
salary
SAT
Read
ing