Expected values

Definition

Expected values are the theoretical frequencies of outcomes in a distribution, calculated based on a specified model. They are used to determine how well observed data fits an expected distribution.

5 Must Know Facts For Your Next Test

1. Expected values are calculated using the formula $E_i = N \times p_i$, where $N$ is the total number of observations and $p_i$ is the probability of each outcome.
2. They play a crucial role in the Chi-Square Goodness-of-Fit test, which compares observed frequencies to expected frequencies.
3. The Chi-Square statistic is computed as $$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$$ where $O_i$ and $E_i$ are observed and expected frequencies, respectively.
4. A significant difference between observed and expected values indicates that the observed data does not fit the expected distribution well.
5. Expected values must be greater than or equal to 5 for each category to ensure the validity of the Chi-Square test.

Review Questions

• What formula is used to calculate expected values?
• How do expected values relate to the Chi-Square Goodness-of-Fit test?
• Why must each expected value be at least 5 for a valid Chi-Square test?

Related terms

Chi-Square Statistic:

$\chi^2$ measures how much observed frequencies deviate from expected frequencies.

Goodness-of-Fit Test:

A statistical test used to determine if sample data matches an expected distribution.

Observed Frequencies:

The actual counts of outcomes obtained from collected data.