Tutoring Use and Math Grades
Of 200 students surveyed at a high school, 80 reported using the after-school tutoring center at least once a month; the other 120 did not. Of the 80 who used tutoring, 56 earned a C or better in their math course that term. Of the 120 who did not use tutoring, 54 earned a C or better. A parent newsletter states: "70% of our students earn a C or better in math, and the tutoring center is clearly the reason."
1. Compute the percentage earning a C or better for each group and for all students overall.
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Tutoring users: $56/80 = 0.70$, or 70%.
Non-users: $54/120 = 0.45$, or 45%.
Overall: $(56 + 54)/200 = 110/200 = 0.55$, or 55%.
The gap between the groups is 25 percentage points. Note that the denominator matters at every step: 70% is a rate among tutoring users only, while 55% is the rate among everyone surveyed.
2. The newsletter claims that 70% of students earn a C or better. Is that supported by the data? Explain where the number came from.
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No. 70% is the conditional rate among tutoring users (56 of 80). The correct overall rate is 55% (110 of 200).
Using 70% as the school-wide figure conflates two very different denominators:
- $56/80$ - share of tutoring users with a C or better.
- $110/200$ - share of all surveyed students with a C or better.
The 70% figure overstates the school-wide result by 15 percentage points, and it also happens to be the exact rate that makes the tutoring program look strongest - a sign the statistic was selected rather than computed.
3. The newsletter also says tutoring "is clearly the reason" for the higher grades. Assess that claim and describe what evidence would support a causal conclusion.
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The claim is not supported. This is an observational comparison, not an experiment, so the 70% vs 45% gap mixes the effect of tutoring with pre-existing differences between the groups.
Plausible confounders:
- Self-selection: students who seek tutoring may already be more motivated, better organized, or under pressure from parents.
- Teacher referral: the center may serve students who were already improving or who have extra support at home.
- Reverse timing: we do not know whether tutoring came before the grade or after a student was already struggling and improving.
The 25-point gap is real as a description of these 200 students, but it is not evidence of a 25-point effect. Stronger evidence would come from randomly assigning some students to the center and comparing grade outcomes, or from a matched comparison that tracks prior math grades and adjusts for them.