In AP Statistics, an expected count is the frequency you would expect in a category if the null hypothesis were true. For a goodness-of-fit test it equals (sample size)(null proportion); in a two-way table it equals (row total)(column total)/table total. Chi-square tests compare these to observed counts.
An expected count answers one question: if the null hypothesis were exactly true, how many observations should land in this category? It's the "no surprise here" baseline. The CED puts it simply: an expected count is a sample size times a probability.
There are two formulas, depending on the test. For a chi-square goodness-of-fit test (Topic 8.2), the null hypothesis hands you the proportions directly, so each expected count is (sample size)(null proportion). For a two-way table (Topic 8.4), the null hypothesis says the two variables are independent, so you build the expected count from the table's margins: expected count = (row total)(column total)/table total. Either way, expected counts are usually decimals, and that's fine. They're theoretical frequencies, not real people or things, so you never round them to whole numbers. Once you have them, the chi-square statistic measures how far the observed counts stray from these expected counts, relative to the expected counts.
Expected counts live in Unit 8 (Inference for Categorical Data: Chi-Square) and show up in three separate learning objectives. AP Stats 8.2.D has you calculate them for a goodness-of-fit test, AP Stats 8.4.A has you calculate them for two-way tables, and AP Stats 8.2.E uses them to verify conditions, since the large counts check requires all expected counts to be greater than 5. That makes expected counts triple-duty knowledge. You need them to set up the test, to check whether the test is even valid, and to compute the chi-square statistic itself. Every chi-square problem on the exam runs through expected counts at some point, whether the question asks for them explicitly or buries them inside a conditions check.
Keep studying AP Statistics Unit 8
Observed Count (Unit 8)
Observed counts are what actually happened in your data; expected counts are what the null hypothesis predicted. The entire chi-square statistic is built from the gap between these two, summed across every cell.
Chi-Square Statistic (Unit 8)
The chi-square statistic is Σ(observed − expected)²/expected. Notice expected counts appear twice, once in the gap and once in the denominator, so a discrepancy of 10 matters way more when you expected 15 than when you expected 500.
Null Hypothesis (Units 6-9)
Expected counts are the null hypothesis turned into numbers. In a goodness-of-fit test the null proportions come straight from H₀; in a two-way table the independence claim in H₀ is what justifies the row-times-column formula.
10% Rule (Units 6-8)
Conditions for chi-square inference pair the 10% condition (n ≤ 10% of N when sampling without replacement) with a large counts check that uses expected counts, not observed ones. All expected counts must exceed 5 for the chi-square approximation to be trustworthy.
Multiple-choice questions love handing you a two-way table's margins and asking for the expected count in one cell, exactly the (row total)(column total)/table total computation from 8.4.A. Watch for stems that say "if the variables are independent," since that phrasing is your cue to compute an expected count. On FRQs, expected counts show up inside full chi-square tests. The 2017 FRQ (age at diagnosis for 207 schizophrenia patients) and the 2019 tumbleweed FRQ both required chi-square inference where expected counts are part of the mechanics and the conditions check. Two grading traps to avoid. First, check the large counts condition with expected counts greater than 5, not observed counts. Second, don't round expected counts to whole numbers; a value like 13.6 is correct as is.
Observed counts come from the data you collected. Expected counts come from a formula and the null hypothesis. If you surveyed 200 students and 90 preferred group study, 90 is observed. If independence predicts that cell should hold 75, then 75 is expected. Mixing these up is fatal in the conditions check, where only expected counts need to be greater than 5, and in the chi-square formula, where expected counts go in the denominator.
An expected count is the count a category would have if the null hypothesis were exactly true, and in general it equals a sample size times a probability.
For a chi-square goodness-of-fit test, each expected count is (sample size)(null proportion), as stated in AP Stats 8.2.D.
For a two-way table, the expected count in any cell is (row total)(column total)/table total, as stated in AP Stats 8.4.A.
The large counts condition for chi-square inference requires every expected count to be greater than 5, and you check expected counts, never observed counts.
Expected counts are usually decimals and should not be rounded to whole numbers, because they're theoretical frequencies rather than actual observations.
The chi-square statistic measures the distance between observed and expected counts relative to expected counts, so expected counts sit in both the numerator and denominator of the formula.
It's the frequency a category would have if the null hypothesis were true. For a goodness-of-fit test it's (sample size)(null proportion); for a two-way table it's (row total)(column total)/table total. It's the baseline you compare your observed data against in a chi-square test.
No. Expected counts are theoretical frequencies, so values like 13.6 or 75.4 are completely normal and you should leave them unrounded. Rounding them to whole numbers actually introduces error into your chi-square statistic.
Observed counts are the actual tallies from your sample. Expected counts are computed from the null hypothesis. The chi-square statistic, Σ(observed − expected)²/expected, exists precisely to measure how far apart these two sets of numbers are.
Expected counts. The conservative check from the CED is that all expected counts must be greater than 5. Checking observed counts instead is a common mistake that costs points on FRQ conditions checks.
Multiply that cell's row total by its column total, then divide by the table total. So if a row total is 120, a column total is 75, and the table has 200 observations, the expected count is (120)(75)/200 = 45. This formula comes from assuming the two variables are independent.
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