Calculating Expected Counts from Row Totals, Column Totals, and the Table Total
Video Follow-Along Worksheet
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Learning Objectives:
Calculate expected counts in a two-way table using row totals, column totals, and the table total.
Explain that expected counts in a two-way table are based on the assumption of no relationship between the variables.
Use the expected-count formula for specific cells in a two-way table.
Use row totals, column totals, and subtraction to find remaining expected counts efficiently.
Distinguish observed counts, expected counts, and total cells in a two-way table.
Essential Knowledge:
A two-way table summarizes data for two categorical variables.
Expected counts in a two-way table are calculated assuming there is no relationship between the row and column variables.
Before finding expected counts, you need the row totals, the column totals, and the table total.
The expected count for any cell is (row total × column total) / table total.
The formula can be understood by applying an overall row proportion to a specific column total.
Once some expected counts are known, the remaining interior cells can often be found by subtraction.
Expected counts are found only for the interior cells of the table, not for the totals.
If expected counts are shown in parentheses, a clear key should identify them.
Key Vocabulary
Two-Way Table
A table that summarizes counts for two categorical variables.
Expected Count
The count predicted for a cell if there is no relationship between the two variables.
Observed Count
The actual count recorded in a cell of the table.
Row Total
The total count across one row of the table.
Column Total
The total count down one column of the table.
Table Total
The grand total of all counts in the table.
No Relationship
The assumption that the row and column variables are not associated.
Cell
A single interior box in a two-way table where a row and column meet.
Subtraction Shortcut
Using row totals and column totals to find remaining expected counts after some are calculated.
Key
A label that explains what numbers in parentheses represent.
Video 1: Expected Counts in Two-Way Tables
[0:00 – 7:51]
1.[0:23]
The main question for this lesson is how to calculate in two-way tables of categorical data.
2.[1:07]
A two-way table summarizes data for categorical variables.
Context — Where Do You Go To School?[1:30 – 1:49]
A random sample of parents with school-aged children was taken in 2019, and a separate random sample was taken in 2020. Parents were asked what type of school their children attended.
School Type
2019 Sample
2020 Sample
Row Total
Public
266
163
429
Private/Parochial/Charter
16
21
37
Home
38
30
68
Column Total
320
214
534
3.[2:01]
Expected counts in a two-way table are calculated assuming there is no between the two categorical variables.
4.[2:22]
The first totals to compute are the .
5.[2:31]
Next, compute the for each year.
6.[2:40]
The table total for the school example is .
7.[3:26]
Because 429/534 = 0.8034, about of the 2019 sample would be expected to attend public school if there is no relationship.
8.[3:51]
The expected count for Public in 2019 is .
9.[4:10]
The general expected-count formula divides by the .
Expected Count Rule[4:10 – 6:00]
Expected count = (row total × column total) / table total
Find the row totals, column totals, and the table total.
Use the formula for one or two cells to get started.
Use subtraction with row totals and column totals to find remaining interior cells when possible.
Do not calculate expected counts for the totals row or totals column.
10.[4:38]
The expected count for Private/Parochial/Charter in 2019 is .
11.[5:29]
Using subtraction from the 2019 column total, the expected count for Home in 2019 is .
12.[5:45]
Using subtraction from the Public row total, the expected count for Public in 2020 is .
13.[6:00]
The expected count for Private/Parochial/Charter in 2020 is .
14.[6:00]
The expected count for Home in 2020 is .
15.[6:00]
We do not calculate expected counts for the .
16.[6:00]
If expected counts are written in parentheses, include a clear to identify them.
Practice Context — Are You Working?[6:35 – 7:02]
A random sample of 2,000 adults recorded education level and employment status. Education level is shown in the columns, and employment status is shown in the rows.
Employment Status
No HS Diploma
HS Diploma No College
HS Diploma Some College
Row Total
Employed
206
548
1186
1940
Unemployed
14
22
24
60
Column Total
220
570
1210
2000
17.[6:35]
In the practice table, education level is represented by the .
18.[6:35]
Employment status is represented by the .
19.[7:02]
The expected count for Employed and No HS Diploma is .
20.[7:02]
The expected count for Employed and HS Diploma / No College is .
21.[7:02]
After calculating a few expected counts with the formula, the remaining cells can often be found by .
Key Takeaways
[Summary]
Expected counts in a two-way table are based on the assumption of no relationship between the two categorical variables.
Always start with the row totals, column totals, and table total.
Use expected count = (row total × column total) / table total.
Use subtraction to finish the table once enough expected counts are known.
Do not compute expected counts for the totals, and label parentheses with a clear key if expected counts are shown in the table.
Reflection Questions
[AI-Graded]
R1.Explain how the school-type-by-year table is used to calculate expected counts. Include the assumption of no relationship, the row totals and column totals, the table total 534, the formula (row total × column total) / table total, the expected count 257.1 for Public in 2019, and how subtraction gives the remaining expected counts.
R2.Explain how the employment example uses both the expected-count formula and the subtraction shortcut. Include the totals 1940, 60, 220, 570, 1210, and 2000, the expected counts 213.4 and 552.9 for the first two employed cells, the remaining expected counts 1173.7, 6.6, 17.1, and 36.3, and why totals do not get expected counts.
Exit Ticket
A random sample of 2,000 adults produced the following observed counts for employment status by education level: Employed = 206, 548, 1186 and Unemployed = 14, 22, 24. The column totals are 220, 570, and 1210.
(a) Find the two row totals and the table total.
(b) State the assumption used to calculate expected counts in this two-way table.
(c) Use the formula to find the expected counts for Employed/No HS Diploma and Employed/HS Diploma No College.
(d) Use subtraction to find the remaining four expected counts.
(e) Explain why the totals do not get expected counts and why a key is useful if expected counts are written in parentheses.
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