karenrochon.files.wordpress.com · Web viewIn response to recent controversy over equity in public school expenditures, the Digest of Education Statistics conducted a study focusing

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


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.


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


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%


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


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


Northwest

528.692 ± 16.369 95%

Midwest 576.5 ± 17.037 95%

South526.76

5 ± 14.314 95%Northeas

t 504 ± 19.673 95%


Northwe

st

r 4 Midwest SigMidwe

st


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


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%


Northwe

st

r 4Midwes

t Midwe

st


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

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karenrochon.files.wordpress.com · Web viewIn response to recent controversy over equity in public school expenditures, the Digest of Education Statistics conducted a study focusing