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    The Association Between Quantity of Sleep and Final Exam Performance Among Phoenix Country Day School Students

    Below is my exploration of sleep and grade data for Fall Final Exam week at my local high school, Phoenix Country Day School. I sought to determine whether there is any association, positive or negative, between hours slept before a final exam and final exam performance.

    While there were many lurking variables, I tried to account for as many as possible through careful survey design and random data collection. This project was done as part of my AP Statistics class in addition to being entered in the American Statistics Association's Project Competition. 

    After taking a look at close to 280 responses over five different final exam days, including variables such as caffeine consumption and average hours of sleep received on a normal weeknight, I found some interesting conclusions.

    Take a look at my report (embedded below, be sure to view if full screen), you may be surprised by the results, I certainly was.


    Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 1 The Association Between Quantity of Sleep and Final Exam Performance Among Phoenix Country Day School Students Michael Silverman Mentor: Peter Flanagan-Hyde (AP Statistics Teacher at Phoenix Country Day School) Phoenix Country Day School 3901 E. Stanford Drive Paradise Valley, Arizona, 8525 TABLE OF CONTENTS Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 2 Introduction and Background Information.........................................................................3 Purpose and Hypothesis...................................................................................................3 Materials and Methods.....................................................................................................4 Discussion and Results....................................................................................................6 Conclusion......................................................................................................................16 Appendix (Questionnaire)...............................................................................................17 INTRODUCTION Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 3 Final exams are often regarded as the most stressful period of the semester for students in high school, college, or graduate schools. This time period in students’ lives is often characterized by extreme sleep deprivation and greater variation from a normal sleep cycle than is typical or optimal. While many students often plan to prepare well in advance of their final exams, reality often dictates that they participate in late-night/early morning cramming sessions the evening before their exam in order to be prepared for their exam. These late night study sessions often reduce the amount of sleep students are able to get from the recommended 6-10 hours to 3-4 hours, sometimes 0. Sleep deprivation has many negative effects, ranging from headaches to a decreased ability to concentrate on cognitive-intensive tasks. PURPOSE AND HYPOTHESES As both the scientific community and the American public become more aware of their sleep habits, more questions are being asked in the hopes of determining what, if any, short- and long-term effects sleep deprivation has on us, both as individuals and as collective communities. In this study, I seek to determine whether there is any association among sleep deprivation and final exam performance during Phoenix Country Day School’s first semester final exam week in the high school. There are four primary hypotheses for this project: 1. That Phoenix Country Day School students, on average, do not receive the recommended 8 hours of sleep on an average school night. 2. That Phoenix Country Day School students, on average, do not receive the recommended 8 hours of sleep the night before their final exam during final exam week. Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 4 3. That a greater amount of sleep the night before a final exam is associated with higher final exam performance (e.g. final exam grade). 4. That the consumption of caffeine prior to taking a final exam is associated with a higher final exam grade. MATERIALS AND METHODS 1. Questionnaire Design The questionnaire was designed to be filled out as quickly as possible and is therefore extremely brief at only eight questions. A copy of the questionnaire is included in the appendix of this project. It includes an identifying question asking for the student identification number. This allowed for survey and final exam grade to be matched up at a later date. Other questions considered how many hours of sleep the student regularly receives on an average school night, how many hours of sleep the student received the night before the exam, how long the student studied the night before the exam, whether they studied before the day prior to their exam, and whether they had studied the morning of their exam. Another question asked the student whether he/she had consumed any energy drink(s) or coffee within the 24 hours prior to their exam in order to screen for the potentially confounding variable of caffeine in and/or other stimulants contained in beverages that may or may not affect test performance (This should, and has, been examined in other studies.). The last question provided space for the student to mention anything else they would like, including but not limited to information about their study habits and techniques. 2. Data Collection All students taking the survey were from grades 9-12. There were approximately 250 potential participants on every final exam day. As the survey was entirely Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 5 voluntary, student participation percentage average 22.32% over the five final exam days upon which the surveys were handed out. A student was either able to take a survey as they handed their test to their teacher or as they exited the exam hall. While I cannot detail any observations of how frequently faculty members actually handed the surveys out, I can remark that the rate of handing the surveys out remains relatively constant over the five day final exam period as participation percentages remain similar. Specific survey collection numbers along with their respective valid survey percentages can be found below in Table 1. TABLE 1 – Data Collection Total # of Surveys Collected On Day 58 47 74 51 Total Collected = 279 # Discarded Due To Bad Data 7 6 8 6 6 33 (11.83%) # Discarded Due to No ID Number 16 10 20 11 14 71 (25.45%) # of Surveys Kept for Statistical Significance 38 (65.52%) 34 (72.34%) 46 (62.16%) 32 (65.31%) 31 (60.78%) 175 (62.72%) Day/Exam 12/11/English 12/12/Science 12/15/History 12/17/Math Totals 12/16/Language 49 3. Statistical Methods All analyses were conducted with SAS JMP 7.0, and alpha of 0.05 was utilized to determine statistical significance of data. 4. Grade Methodology and Calculations Grades were originally received in letter grade form as can be seen in the top half of the table below and were converted to point scores, for ease of analysis, according to PCDS’s own GPA calculation system. Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 6 F 0 D.67 D 1 D+ 1.33 C1.67 C 2 C 2.33 B2.67 B 3 B+ 3.33 A3.67 A 4 DISCUSSION AND RESULTS Of 1250 possible student responses (250 students per final exam X 5 final exams), there were 279 (22.32%) responses collected. Of those 279, 33 (11.83%) were discarded due to “bad,” or impossible data (e.g. the hours they reported were inconsistent with the numbers of hours in a single day). In addition, of the 279 total responses, 71 (25.45%), were discarded due to the absence of a student identification number. This is a key part of the survey, as without it, final exam grades cannot be correlated with survey results. In total, 175 (62.72%) survey results presented data that was reasonable and identifiable. Analysis of Average # of Hours on a Regular School Night Average Hours Sleep School Night – With Outliers 3 4 5 6 7 8 9 10 Quantiles 100.0% maximum 75.0% quartile 50.0% median 25.0% quartile 0.0% minimum Moments Mean Std Dev Std Err Mean upper 95% Mean 9.5000 8.0000 7.0000 6.8750 3.0000 7.2456897 1.0979731 0.0832371 7.4099807 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 7 lower 95% Mean N 7.0813986 175 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 8 Confidence Intervals – 95% Parameter Mean Estimate 7.24569 Lower CI 7.081399 Lower CI 7.028895 Upper CI 7.409981 Upper CI 7.462485 1-Alpha 0.950 1-Alpha 0.990 Confidence Intervals – 99% Parameter Estimate Mean 7.24569 Test Mean Hypothesized Value Actual Estimate df Std Dev Test Statistic Prob > |t| Prob > t Prob < t 8 7.24569 173 1.09797 t Test -9.0622 <.0001 1.0000 <.0001 7.0 7.5 8.0 8.5 9.0 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 9 In response to the first hypothesis, that Phoenix Country Day School students, on average, do not receive the recommended 8 hours of sleep on an average school night, one can see in the above tables statistical evidence that indicates that the average PCDS student receives less than the recommended 8 hours of sleep on an average school night. In a 99% confidence interval, students are predicted to have received between 7.03 and 7.46 hours of sleep per night, with a mean of 7.25 hours, a little more than half an hour less than the recommended number of hours of sleep per night. Approximately 40.26%(Green in Pie Chart) of students received 8 hours of sleep or more on an average school night. In a two sided significance test, this amount is statistically significant with a p-value <.0001. Even if one removes the outliers seen on the left side of the histogram, the results are still statistically significant, with a p-value <.0001. Again, if outliers are removed, the histogram indicates that the distribution of hours of sleep received by PCDS students is approximately normal, even though it is slightly skewed to the right. Test Mean=value Hypothesized Value Actual Estimate df Std Dev Test Statistic Prob > |t| Prob > t Prob < t <.0001 8 7.41921 163 0.85171 t Test -8.7327 <.0001 1.0000 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 10 2.4% 5.3%0.6% 0.6% 1.8% 1.8% 8.3% 6.5% 9.5% N 26.6% 27.2% 5 6 7 8 9 10 0.6% 8.9% Average Hours Sleep School Night Average Hours Sleep School Night 3 7.5 4 7.75 5 8 5.5 8.5 6 9 6.5 9.5 7 Analysis of Average # of Hours of Sleep The Night Before a Final Exam Average Hours Sleep Pre-Final Exam Night – With Outliers N Test Mean=value Hypothesized Value Actual Estimate df Std Dev 0 1 2 3 4 5 6 7 8 9 10 11 175 8 7.24857 174 1.68826 Quantiles 100.0% 75.0% 50.0% 25.0% 0.0% Moments maximum quartile median quartile minimum 10.000 8.500 7.500 6.500 1.000 Test Statistic Prob > |t| Prob > t Prob < t t Test -5.8880 <.0001 1.0000 <.0001 Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean 7.2485714 1.6882575 0.1276203 7.5004545 6.9966884 7.0 7.5 8.0 8.5 9.0 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 11 Confidence Intervals Parameter Estimate Mean 7.248571 Std Dev 1.688258 Lower CI 6.9162 1.482017 Upper CI 7.580943 1.955391 1-Alpha 0.990 In response to the second hypothesis, that Phoenix Country Day School students, on average, do not receive the recommended 8 hours of sleep on the night before a final exam, one can see in the above tables statistical evidence that indicates that the average PCDS student receives less than the recommended 8 hours of sleep on the night before their exam. In a 99% confidence interval, students are predicted to have received between 6.92 and 7.58 hours of sleep per night, with a mean of 7.25 hours, a little less than half an hour less than the recommended number of hours of sleep per night. Interestingly, while the mean # of hours of sleep is approximately the same, both the confidence interval and standard deviation indicate a greater amount of variation in the sleep received before an exam. Perhaps most striking is the change in standard deviation from 1.0979731 to 1.68826. Approximately 61.97%(Green in Pie Chart on Next Page) of students received 8 hours of sleep or more on the night before their exam. In a two sided significance test, this amount is statistically significant with a p-value <.0001. Even if one removes the outliers seen on the left side of the histogram, the results are still statistically significant, with a p-value <.0001. Again, if outliers are removed, the histogram indicates that the distribution of hours of sleep received by PCDS students is approximately normal.. Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 12 lower 95% Mean N 7.5035194 156 Test Mean=value Hypothesized Value Actual Estimate df Std Dev Test Statistic Prob > |t| Prob > t Prob < t t Test -3.4021 0.0009 0.9996 0.0004 8 7.6859 155 1.15314 5 6 7 8 9 10 Normal(7.6859,1.15314) Quantiles 100.0% 75.0% 50.0% 25.0% 0.0% maximum quartile median quartile minimum 10.000 8.500 8.000 7.000 5.000 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 Moments Mean Std Dev Std Err Mean upper 95% Mean 7.6858974 1.1531415 0.0923252 7.8682755 Confidence Intervals Parameter Mean Std Dev Estimate Lower CI 7.685897 7.445121 1.153141 1.004915 Upper CI 7.926674 1.348133 1-Alpha 0.990 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 13 3.96% 4.17% 2.09% 0.46% 8.01% 2.71% % of Total(Hours Sleep Night Before Exam) 17.26% 21.02% 9.22% 3.75% 27.36% Hours Sleep Night Before Exam Hours Sleep Night Before Exam 5 10 5.5 6 6.5 7 7.5 8 8.5 9 9.5 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 14 Analysis of Association among Hours Sleep Night Before Final Exam and Final Exam Grade – With Outliers 4 3 Grade on Exam 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 Hours Sleep Night Before Exam Linear Fit Linear Fit Grade on Exam = 2.5563936 + 0.0827758*Hours Sleep Night Before Exam Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 0.026597 0.020971 0.847854 3.1564 175 Analysis of Variance Source Model Error C. Total DF 1 173 174 Sum of Squares 3.39808 124.36215 127.76023 Mean Square 3.39808 0.71886 F Ratio 4.7271 Prob > F 0.0310 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 15 Parameter Estimates Term Intercept Hours Sleep Night Before Exam Estimate 2.5563936 0.0827758 Std Error 0.283314 0.038072 t Ratio 9.02 2.17 Prob>|t| <.0001 0.0310 In response to the third hypothesis, that a greater amount of sleep the night before a final exam is associated with higher final exam performance (e.g. final exam grade) among PCDS upper school students, on can see on the previous page there is a positive association between the hours a student sleeps before their final exam and their performance on said exam as the p-value is .0310*. However, while their may be statistical significance in regards to test performance and the number of hours of sleep before the exam, is this statistical significance of practical significance? Hours of Sleep 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Grade Via Linear Fit 2.639169 4000 2.721945 2000 2.804721 0000 2.887496 8000 2.970272 6000 3.053048 4000 3.135824 2000 3.218600 0000 3.301375 8000 3.384151 6000 3.466927 4000 3.549703 2000 3.632479 0000 3.715254 8000 Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 16 15 16 17 18 3.798030 6000 3.880806 4000 3.963582 2000 4.046358 0000 It is not necessarily of practical significance, as the R-Square value is extremely low, indicating that the linear line of fit is not a very accurate model. In addition, if one examines the table at the left, one can see that, to attain an exam grade of “A,” one would necessarily need to sleep approximately 17.5 hours. If one examines the other end of the spectrum, even if one receives only 1 hour of sleep, one should be able to attain an approximate grade of B-. In addition, there are other lurking variables that influence what grade a student receives on an exam. For example, if one examines the following distributions for grades if students drank a caffeinated beverage before or of the day of the test. The grade distribution for students who did drink caffeinated beverages are on the left, while the grade distribution for those that did not drink caffeinated beverages before their exam are on the right. Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 17 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 Quantiles 100.0% 75.0% 50.0% 25.0% 0.0% maximum quartile median quartile minimum 4.0000 4.0000 3.0000 2.6700 0.6700 Quantiles 100.0% 75.0% 50.0% 25.0% 0.0% maximum quartile median quartile minimum 4.0000 4.0000 3.3300 2.6700 0.0000 Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N 3 0.9428735 0.1509806 3.3056442 2.6943558 39 Upper CI 3.305644 1.215155 1-Alpha 0.950 Moments Mean Std Dev Std Err Mean upper 95% Mean lower 95% Mean N 3.20125 0.8288783 0.0710757 3.3418159 3.0606841 136 Estimate 3.20125 0.828878 Lower CI 3.060684 0.7407 Upper CI 3.341816 0.941077 1-Alpha 0.950 Confidence Intervals Parameter Mean Std Dev Lower CI 3 2.69435 6 0.942874 0.77056 Estimate Confidence Intervals Parameter Mean Std Dev Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 18 In what may seem contradictory to common knowledge, both the mean and the median final grade exam of the students that did not drink any caffeinated beverages before their exam was higher, 3 versus 3.2 for the mean, and 3 versus 3.33 for the median. In addition, there is much great variation in the grade distribution of those who drank caffeinated beverages and than in the distribution of those who did not. While this may likely be caused by fewer trials, it is still interesting to note. 1.5% 0.7% 4.4% 5.1% 31.6% 8.1% 8.8% No Proportionally, some interesting trends appear when comparing the grades of caffeinated versus noncaffeinated test takers. Over 30% of “non-caffeinated” test N 14.7% 11.0% 14.0% Grade on Exam Column 9 takers received a grade of “A” while only 25.6% of caffeinated students did. In addition, the 25.6% 2.6% 5.1% 5.1% 5.1% 2.6% 10.3% percentage of “non-caffeinated” students who receive a grade of “A-“is almost twice that of “caffeinated” students. Therefore, in response to hypothesis four, it appears that Yes N 7.7% 12.8% 23.1% Grade on Exam there is a negative association between final exam grade and Grade on Exam 0 2.67 0.67 3 1 3.33 1.33 3.67 1.67 4 2 2.33 caffeine consumption. Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 19 CONCLUSION The results found in this study indicate that while hours slept before a final exam and final exam performance are positively associated, that association is not practically significant. In addition, PCDS students, on average, do not received the recommended eight hours of sleep per night, regardless of whether it is before a final exam or on an average school night. However, perhaps the most surprising results of this study were in the grade distributions of “caffeinated” and “non-caffeinated” test takers; that there is in fact a negative association between the two. Students can conclude from this study that the key to successful performance on final exams is a reasonable amount of sleep that is likely specific to each individual. Students should also refrain from ingesting to much caffeine, as test performance is may suffer. Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 20 Thank you for your participation in this AP Statistics Project. Please be as honest and specific as possible. All data is anonymously collected, your name will never be used. Student ID Number -- _______________________________ 1. How much sleep do you get on an average school night? Answer in hours ___________________________________________ 2. How many hours of sleep did you get last night? Answer in hours ___________________________________________ 3. For how long did you study yesterday/last night? Answer in hours ____________________________________________ 4. Did you begin studying for this exam before yesterday? Answer (Yes/No, Which Days) ________________________________ 5. What time did you study until last night? Answer (Time, pm/am) ______________________________________ 6. Did you study this morning? Answer (Yes/no, how long) ___________________________________ 7. Did you consume an energy drink(s)/coffee any time within the last 24hrs? If so, which one/ how many? Be as specific if you can. Answer - _______________________________________________ 8. If you would like to mention anything else, something about your study habits, please answer below. Answer _______________________________________________________ ________________________________________________________ _______ Michael Silverman, Phoenix Country Day School, ASA Statistics Project Competition, 21