What is Expected Frequencies in Chi-Square?

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Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Expected frequency in Chi-Square is the number of times we would *expect* an event to happen if there were no special relationship or pattern between the things we are studying. It's like a prediction based on pure chance or what's 'normal'. We use it to compare against what we actually observe.

Simple Example

Quick Example

Imagine you have 100 students, and you want to see if boys and girls like cricket equally. If there are 50 boys and 50 girls, and 60 students like cricket overall, we would *expect* 30 boys (50/100 * 60) and 30 girls (50/100 * 60) to like cricket, assuming no difference between genders. These 30s are the expected frequencies.

Worked Example

Step-by-Step

Let's say a snack company wants to know if people in different cities prefer their new 'Masala Munch' flavour. They survey 200 people:

Observed Data:
| City | Like Masala Munch | Don't Like Masala Munch | Total |
|----------|-------------------|-------------------------|-------|
| Delhi | 40 | 30 | 70 |
| Mumbai | 60 | 70 | 130 |
| Total | 100 | 100 | 200 |

We need to calculate the expected frequency for each cell.

--- Step 1: Calculate Row and Column Totals. (Already done in the table above).
Row Totals: Delhi = 70, Mumbai = 130.
Column Totals: Like Masala Munch = 100, Don't Like Masala Munch = 100.
Grand Total = 200.

--- Step 2: Use the formula: Expected Frequency = (Row Total * Column Total) / Grand Total.

--- Step 3: Calculate Expected Frequency for 'Delhi' and 'Like Masala Munch'.
Expected = (Row Total for Delhi * Column Total for Like Masala Munch) / Grand Total
Expected = (70 * 100) / 200 = 7000 / 200 = 35

--- Step 4: Calculate Expected Frequency for 'Delhi' and 'Don't Like Masala Munch'.
Expected = (Row Total for Delhi * Column Total for Don't Like Masala Munch) / Grand Total
Expected = (70 * 100) / 200 = 7000 / 200 = 35

--- Step 5: Calculate Expected Frequency for 'Mumbai' and 'Like Masala Munch'.
Expected = (Row Total for Mumbai * Column Total for Like Masala Munch) / Grand Total
Expected = (130 * 100) / 200 = 13000 / 200 = 65

--- Step 6: Calculate Expected Frequency for 'Mumbai' and 'Don't Like Masala Munch'.
Expected = (Row Total for Mumbai * Column Total for Don't Like Masala Munch) / Grand Total
Expected = (130 * 100) / 200 = 13000 / 200 = 65

--- Expected Frequencies Table:
| City | Like Masala Munch | Don't Like Masala Munch | Total |
|----------|-------------------|-------------------------|-------|
| Delhi | 35 | 35 | 70 |
| Mumbai | 65 | 65 | 130 |
| Total | 100 | 100 | 200 |

Answer: The expected frequencies are 35 for Delhi/Like, 35 for Delhi/Don't Like, 65 for Mumbai/Like, and 65 for Mumbai/Don't Like.

Why It Matters

Understanding expected frequencies helps scientists, economists, and data analysts see if observed results are just random or if there's a real reason behind them. In AI/ML, it helps evaluate model performance, and in physics, it can be used to test theories against experimental data. People working in market research, medical trials, or even sports analytics use this to make informed decisions and discover hidden patterns.

Common Mistakes

MISTAKE: Confusing expected frequency with observed frequency. | CORRECTION: Observed frequency is what actually happened or was counted. Expected frequency is what we *predict* would happen if there's no relationship between the variables.

MISTAKE: Using only row totals or column totals in the calculation, instead of both. | CORRECTION: The formula for expected frequency is (Row Total * Column Total) / Grand Total. All three components are crucial.

MISTAKE: Calculating expected frequencies for percentages instead of actual counts. | CORRECTION: Chi-Square calculations, including expected frequencies, must always be done using raw counts or actual numbers, not percentages or ratios.

Practice Questions

Try It Yourself

QUESTION: A class of 80 students has 40 boys and 40 girls. If 60 students passed an exam, what is the expected number of boys who passed, assuming no difference between boys and girls? | ANSWER: Expected Boys Passed = (Total Boys * Total Passed) / Grand Total = (40 * 60) / 80 = 2400 / 80 = 30 boys.

QUESTION: In a survey of 150 people about their favourite fruit, 90 preferred mangoes and 60 preferred apples. If 75 of the respondents were from a rural area and 75 from an urban area, what is the expected number of urban people who prefer mangoes? | ANSWER: Expected Urban Mango Lovers = (Total Urban * Total Mango Lovers) / Grand Total = (75 * 90) / 150 = 6750 / 150 = 45 urban people.

QUESTION: A company launched two new phone models, A and B. Out of 500 customers, 300 bought Model A and 200 bought Model B. If 250 customers were under 30 years old and 250 were 30 or older, calculate the expected number of customers under 30 who bought Model B. | ANSWER: Expected (Under 30, Model B) = (Total Under 30 * Total Model B) / Grand Total = (250 * 200) / 500 = 50000 / 500 = 100 customers.

MCQ

Quick Quiz

What does an expected frequency represent in a Chi-Square test?

The actual number of observations in a category.

The number of observations we would predict if there's no relationship between variables.

The difference between observed and actual frequencies.

The total sum of all observations.

The Correct Answer Is:

Expected frequency is a theoretical value that tells us what we would anticipate seeing in each category if the variables were independent (i.e., had no relationship). Option A is the observed frequency. Options C and D are incorrect definitions.

Real World Connection

In the Real World

When you see news reports about election poll results, analysts often use expected frequencies to check if certain demographic groups (like young voters vs. old voters, or people from different states) voted differently than expected by chance. This helps them understand voting patterns and predict outcomes. Similarly, in medical research, it's used to see if a new medicine works better than a placebo, by comparing actual patient recovery rates to what would be expected if the medicine had no effect.

Key Vocabulary

Key Terms

OBSERVED FREQUENCY: The actual count of how many times something happened. | CHI-SQUARE TEST: A statistical test to see if there's a significant relationship between two categorical variables. | CATEGORICAL VARIABLE: A variable that can be divided into groups or categories (e.g., gender, city, phone model). | INDEPENDENCE: When two variables have no effect on each other.

What's Next

What to Learn Next

Great job understanding expected frequencies! The next step is to learn how to calculate the Chi-Square statistic itself. This builds directly on expected frequencies, as you'll use both observed and expected values to determine if the differences you see are statistically significant or just due to chance. Keep up the amazing work!