Introduction and Rationale
Students are often told that the key to a high grade is to study more, yet some students who spend long hours with their notes earn modest marks while others who study less perform very well. This raises a fair statistical question: is study time a strong indicator of exam performance, or do prior knowledge, concentration and technique matter more? Research conducted by Macnamara and Burgoyne (2023) has shown that there is no clear relationship between amount of study and academic performance. This is because the number of studies varies from one student to another. Thus, study hour alone does not explain fully the relationship and thus there is a need to measure it.
This report examines the relationship for a small group of students who reported the hours they spent studying for a mathematics exam and the score they obtained. It uses a scatter plot, the Pearson correlation coefficient and a simple linear regression. The research question is: To what extent does the number of hours students study correlate with their exam scores?
Aim and Hypotheses
The aim is to quantify the correlation between weekly study hours and math exam scores and to describe the relationship with a regression line. With the population correlation written ρ, the null hypothesis is H₀: ρ = 0 (no linear relationship) and the alternative is H₁: ρ ≠ 0. A two-tailed test was used, although a positive association was expected. The significance level was α = .05.
Method
Fifteen students were asked how many hours they spent studying for their mathematics exam and what score (out of 100) they received. Study hours are the explanatory variable (X) and exam score is the response variable (Y). Both are continuous and measured on a ratio scale, so Pearson correlation is appropriate. The raw data are listed in Table 1 in the Raw Data section. The analysis had four steps: descriptive statistics, a scatter plot to check linearity and unusual points, the Pearson coefficient (calculated by hand from the deviation table, confirmed with software and tested with a t statistic on n − 2 degrees of freedom), and a least squares regression line. The Pearson coefficient is the sum of the products of the deviations from the means divided by the square root of the product of the two sums of squared deviations (Chattamvelli, 2024):
Here is a student’s study hours, the exam score, and and the sample means. The coefficient ranges from −1 to +1; values near either end indicate a strong linear association and values near zero indicate little or none (Okoye & Hosseini, 2024).
Raw Data
Table 1 lists the data collected from the 15 students: x is the number of hours studied and y is the exam score out of 100. The totals at the bottom are used in the calculations that follow.
| Student | x (hours) | y (score) |
|---|---|---|
| 1 | 1 | 50 |
| 2 | 2 | 55 |
| 3 | 3 | 60 |
| 4 | 4 | 65 |
| 5 | 4 | 68 |
| 6 | 5 | 70 |
| 7 | 5 | 72 |
| 8 | 6 | 75 |
| 9 | 6 | 78 |
| 10 | 7 | 80 |
| 11 | 7 | 82 |
| 12 | 8 | 85 |
| 13 | 8 | 88 |
| 14 | 9 | 90 |
| 15 | 10 | 95 |
| Total | 85 | 1113 |
Note. x = weekly study hours; y = exam score (out of 100). Student numbers are labels only.
Calculations and Analysis
Means and descriptive statistics
The means come from the totals in Table 1, with n = 15 students.
Students studied an average of 5.67 hours (SD = 2.58), with values from 1 to 10 hours, and their scores averaged 74.20 (SD = 13.09), with a range from 50 to 95 (Table 2). The mean and median of each variable are close, which suggests no serious skew.
| Variable | M | SD | Median | Min | Max |
|---|---|---|---|---|---|
| Study hours | 5.67 | 2.58 | 6.00 | 1 | 10 |
| Exam score | 74.20 | 13.09 | 75.00 | 50 | 95 |
Figure 1 below shows the scatter plot. The points rise steadily from lower left to upper right and lie very close to a straight line, with no curvature and no outlier, so a linear model is suitable.
- One student
- Least squares line: Ŷ = 45.60 + 5.05X, r = .995, R² = .990
Point at a student, or use the number buttons (arrow keys move between them), to see the score the line predicts and the residual.
Student 5
- Weekly study hours
- 4
- Math exam score
- 68
- Predicted score
- 65.79
- Residual
- 68 − 65.79 = +2.21: 2.21 above the line
Figure 1 data: each student’s hours, score, predicted score and residual
| Student | Weekly study hours | Math exam score | Predicted score | Residual |
|---|---|---|---|---|
| 1 | 1 | 50 | 50.65 | −0.65 |
| 2 | 2 | 55 | 55.70 | −0.70 |
| 3 | 3 | 60 | 60.74 | −0.74 |
| 4 | 4 | 65 | 65.79 | −0.79 |
| 5 | 4 | 68 | 65.79 | +2.21 |
| 6 | 5 | 70 | 70.84 | −0.84 |
| 7 | 5 | 72 | 70.84 | +1.16 |
| 8 | 6 | 75 | 75.88 | −0.88 |
| 9 | 6 | 78 | 75.88 | +2.12 |
| 10 | 7 | 80 | 80.93 | −0.93 |
| 11 | 7 | 82 | 80.93 | +1.07 |
| 12 | 8 | 85 | 85.97 | −0.97 |
| 13 | 8 | 88 | 85.97 | +2.03 |
| 14 | 9 | 90 | 91.02 | −1.02 |
| 15 | 10 | 95 | 96.07 | −1.07 |
Checking assumptions
The scatter plot supports linearity, and the spread of points around the line is similar at low and high study hours, so the error variance appears constant. No observation lies far from the others. The Shapiro–Wilk test did not indicate departures from normality for study hours, W = .98, p = .98, or for exam scores, W = .98, p = .98. Each student reported their own hours, so the observations can reasonably be treated as independent.
Deviations and sums of squares
For each student, the mean is subtracted from x and from y, and the deviations are squared and multiplied together (Table 3). Student 1 is worked out below as an example; the other rows follow the same pattern.
| Student | x | y | xᵢ − x̄ | yᵢ − ȳ | (xᵢ − x̄)² | (yᵢ − ȳ)² | (xᵢ − x̄)(yᵢ − ȳ) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 50 | −4.67 | −24.20 | 21.78 | 585.64 | 112.93 |
| 2 | 2 | 55 | −3.67 | −19.20 | 13.44 | 368.64 | 70.40 |
| 3 | 3 | 60 | −2.67 | −14.20 | 7.11 | 201.64 | 37.87 |
| 4 | 4 | 65 | −1.67 | −9.20 | 2.78 | 84.64 | 15.33 |
| 5 | 4 | 68 | −1.67 | −6.20 | 2.78 | 38.44 | 10.33 |
| 6 | 5 | 70 | −0.67 | −4.20 | 0.44 | 17.64 | 2.80 |
| 7 | 5 | 72 | −0.67 | −2.20 | 0.44 | 4.84 | 1.47 |
| 8 | 6 | 75 | 0.33 | 0.80 | 0.11 | 0.64 | 0.27 |
| 9 | 6 | 78 | 0.33 | 3.80 | 0.11 | 14.44 | 1.27 |
| 10 | 7 | 80 | 1.33 | 5.80 | 1.78 | 33.64 | 7.73 |
| 11 | 7 | 82 | 1.33 | 7.80 | 1.78 | 60.84 | 10.40 |
| 12 | 8 | 85 | 2.33 | 10.80 | 5.44 | 116.64 | 25.20 |
| 13 | 8 | 88 | 2.33 | 13.80 | 5.44 | 190.44 | 32.20 |
| 14 | 9 | 90 | 3.33 | 15.80 | 11.11 | 249.64 | 52.67 |
| 15 | 10 | 95 | 4.33 | 20.80 | 18.78 | 432.64 | 90.13 |
| Total | 85 | 1113 | 0.00 | 0.00 | 93.33 | 2400.40 | 471.00 |
Note. Deviations are rounded to two decimal places; sums use the unrounded values.
Adding each column of Table 3 gives the three sums needed for the correlation and the regression line:
Pearson correlation coefficient
The coefficient is the cross-product sum divided by the square root of the product of the two sums of squares:
A value this close to +1 shows a very strong, positive linear relationship. Thus, students who studied more hours scored higher.
Significance test
To test H₀: ρ = 0, the coefficient is converted to a t statistic with n − 2 = 13 degrees of freedom (using r = .9951 and r² = .9902 before rounding):
The correlation was significant, r(13) = .995, t(13) = 36.24, p < .001, 95% CI [.985, .998], so the null hypothesis was rejected. A Spearman rank correlation agreed, rₛ = .996, p < .001, so the result does not depend on an exactly linear form.
Regression line
The slope is the cross-product sum divided by the sum of squared x deviations, and the intercept makes the line pass through the point of means:
Study hours explained 99.0% of the variance in scores (adjusted R² = .989), and the standard error of the estimate was 1.35 points. Table 4 summarizes the model.
| Predictor | b | SE | 95% CI | t(13) | p |
|---|---|---|---|---|---|
| Intercept | 45.60 | 0.86 | [43.74, 47.47] | 52.89 | < .001 |
| Study hours | 5.05 | 0.14 | [4.75, 5.35] | 36.24 | < .001 |
Note. R² = .990; F(1, 13) = 1313.07, p < .001.
Predicted scores
Substituting 3, 6 and 9 hours into the line gives:
These lie inside the observed range of 1 to 10 hours, so they are interpolations. A gain of about five points per hour is large relative to the standard deviation of scores (13.09). The difference between 3 and 9 hours of study is about 30 points, more than two standard deviations. The intercept of 45.60 is the predicted score at zero hours, just outside the observed range, so it is the starting point of the line and not a literal forecast for a student who does no studying.
Residuals
Residuals are the observed scores minus the scores the line predicts. They ranged from −1.07 to 2.21 points, so no student is far from the prediction. The Shapiro–Wilk test nevertheless rejected normality of the residuals, W = .73, p < .001. With only 15 tightly packed residuals the test is sensitive to a few values lying slightly apart, and it does not indicate a pattern that would invalidate the line.
Conclusion and Evaluation
The data show a very strong, positive and statistically significant correlation between study hours and math exam scores, r(13) = .995, p < .001. Each extra hour of study was associated with about five extra points, and study hours accounted for nearly all the variation in scores in this group. The expectation that more study goes with higher scores was supported, and the answer to the research question is that study hours correlate with exam scores to a very high extent in this sample.
Several points qualify this result. First, correlation is not causation: students who study more may also be more motivated or better prepared. Second, the sample has only 15 students and hours were self-reported, which may be inaccurate. Third, a coefficient of .995 is unusually high for educational data, where ability, sleep, anxiety and technique normally add scatter, so the finding should be replicated before it is generalized. Fourth, the residuals were flagged as non-normal, W = .73, p < .001, but all lie within about two points of the line, so the practical effect is small. Finally, the line should not be extended beyond the observed range: at 12 hours it predicts a score above 100, which is impossible, so the relationship must level off.
Strengths include the use of both a graph and a numerical measure, the Spearman check and significance test, and the hand calculation that makes each step visible. Future work should use a larger sample from several schools, record the quality and timing of study as well as its length, and use multiple regression to compare hours with other predictors. The practical message is balanced. This is because in this group more study time went with higher scores, which supports putting in the hours, but hours are only a proxy for learning, and focused, deliberate methods are likely to matter as well.
