Expected Count in AP Statistics

In AP Statistics, an expected count is the number of observations you would predict in each category or cell if the null hypothesis were true. In a two-way table, it is found with (row total × column total) / table total and compared to observed counts in a chi-square test.

What is Expected Count?

An expected count is the count you'd expect to see in a category (or in one cell of a two-way table) if the null hypothesis were exactly true. Think of it as the "nothing interesting is going on" prediction. A chi-square test then asks one question. How far are the real data from that prediction?

For a chi-square test for homogeneity or independence, the null says there's no association between the two categorical variables (or that the distributions are the same across groups). Under that assumption, each cell's expected count is (row total × column total) / table total. For example, if 200 students are classified by year in school and study habits, and 50 are freshmen and 80 have good study habits, the expected count for "freshman with good habits" is (50 × 80) / 200 = 20. Expected counts don't have to be whole numbers. A value like 12.4 is totally fine, and you shouldn't round it to make it look like a real count.

Why Expected Count matters in AP® Statistics

Expected count sits at the center of Topics 3.14 (Setting Up a Chi-Square Test for Homogeneity or Independence) and 3.15 (Carrying Out a Chi-Square Test for Homogeneity or Independence). It does two separate jobs on those topics. During setup, you use it to check the large counts condition, where every expected count must be at least 5 for the chi-square distribution to be a reasonable model. During carrying out the test, it goes into the chi-square statistic, Σ(observed − expected)² / expected. If you get the expected counts wrong, your condition check, test statistic, and p-value are all wrong too. That's why the exam keeps returning to this one calculation.

How Expected Count connects across the course

Observed Count (Topics 3.14-3.15)

Observed counts are what actually happened, and expected counts are what the null hypothesis predicts. The whole chi-square test is a comparison between the two, so each term only makes sense next to the other.

Chi-Square Statistic (Topic 3.15)

The chi-square statistic is basically a scoreboard of how badly the expected counts missed. Each cell adds (O − E)² / E. Dividing by E means a miss of 5 counts a lot more when you only expected 10 than when you expected 100.

Null Hypothesis and Independence (Topics 3.14-3.15)

The row-total-times-column-total formula is really the probability rule for independent events in disguise. If two variables are independent, P(A and B) = P(A) × P(B), and multiplying that by the table total gives you the expected count.

Random Sampling and Conditions (Topic 3.14)

The large counts condition is checked with expected counts, but it's only one of the conditions. You still need random sampling (such as a simple random sample or a randomized experiment), and large expected counts can't rescue data that wasn't collected randomly.

Is Expected Count on the AP® Statistics exam?

Multiple choice. The most common stem hands you a two-way table, or just its margins, and asks for the expected count in one cell. One example is 200 college students classified by year in school and study habits. The fast move is (row total × column total) / grand total. Other questions ask what would invalidate a chi-square test that already produced a statistic and p-value (for instance, χ² = 9.8 with 2 degrees of freedom). An expected count below 5 is a classic correct answer. Watch for distractors that check observed counts instead.

Free response. Released FRQs built around chi-square tests, like 2017 FRQ Q5 (age at diagnosis for a random sample of 207 men and women being treated for schizophrenia) and 2019 FRQ Q4 (the tumbleweed/kochia context), reward a complete inference procedure. On those questions you need to:

• Show how at least one expected count was calculated, or list them all.
• State that all expected counts are at least 5, which checks the large counts condition.
• Use the expected counts in the chi-square statistic and connect the result to a p-value and a conclusion in context.

Just writing "conditions met" without referencing the expected counts usually costs credit.

Expected Count vs Observed Count

Observed counts are the actual numbers in your data table. Expected counts are calculated numbers that describe what the table would look like if the null hypothesis were true. The big mix-up happens on the large counts condition. You check that the expected counts are all at least 5, not the observed ones. A cell can have an observed count of 2 and still be fine if its expected count is 6. Also, observed counts are always whole numbers, but expected counts often aren't.

Key things to remember about Expected Count

• An expected count is the number you would predict in a cell if the null hypothesis were true.

• In a chi-square test for homogeneity or independence, each expected count equals (row total × column total) divided by the table total.

• Expected counts can be decimals, so don't round them to whole numbers.

• The large counts condition requires every expected count, not every observed count, to be at least 5.

• The chi-square statistic adds up (observed − expected)² / expected across all cells, so bigger gaps between observed and expected give a bigger statistic and a smaller p-value.

• On FRQs, show at least one expected count calculation and explicitly state that all expected counts are at least 5.

Frequently asked questions about Expected Count

What is an expected count in AP Stats?

It's the count you'd predict in a category or table cell if the null hypothesis were true. In a two-way table, you find it with (row total × column total) / table total, and it's what observed counts get compared to in a chi-square test.

Do expected counts have to be whole numbers?

No. Expected counts are long-run averages under the null, so values like 13.75 are normal. Keep the decimals when you compute the chi-square statistic.

Do you check the observed or expected counts for the chi-square conditions?

You check the expected counts. Every expected count needs to be at least 5 for the large counts condition, even if some observed counts are smaller than that.

What's the difference between expected count and observed count?

Observed counts are the real data you collected, while expected counts are calculated from the table's totals assuming the null hypothesis is true. The chi-square statistic measures how far the observed counts land from the expected ones.

How do you calculate expected counts for a chi-square test of independence?

Multiply the cell's row total by its column total, then divide by the grand total. For example, with 200 students, a row total of 50, and a column total of 80, the expected count is (50 × 80) / 200 = 20.