Essential Statistical Tables and Formulas for College Statistics

Standard Normal (Z) Tables

Negative and Positive Z-Tables

The z-table provides cumulative probabilities for the standard normal distribution, which is a normal distribution with mean 0 and standard deviation 1. Z-tables are used to find the probability that a statistic is less than a given z-score.

• Negative z-table: Used for z-scores less than 0 (left of the mean).

• Positive z-table: Used for z-scores greater than 0 (right of the mean).

How to use: Locate the row for the z-score's first two digits and the column for the second decimal place. The intersection gives the cumulative probability.

Z	0.00	0.01	...
-1.0	0.1587	0.1562	...
0.0	0.5000	0.5040	...
1.0	0.8413	0.8438	...
...	...	...	...

Additional info: The full tables provide probabilities for z-scores from approximately -3.4 to +3.4.

Z-Score Formula

• Individual value: $z = \frac{x - \mu}{\sigma}$

• Sample mean (Central Limit Theorem): $z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$

Example: If $x = 85$, $\mu = 80$, $\sigma = 5$, then $z = (85-80)/5 = 1.0$.

T-Distribution Critical Values

T-Table

The t-distribution is used instead of the normal distribution when the sample size is small and/or the population standard deviation is unknown. The t-table provides critical values for various degrees of freedom (df) and significance levels (α).

df	0.10	0.05	0.025	0.01	0.005
1	3.078	6.314	12.706	31.821	63.657
5	1.476	2.015	2.571	3.365	4.032
10	1.372	1.812	2.228	2.764	3.169
...	...	...	...	...

Additional info: Use the row for your sample's degrees of freedom and the column for your desired significance level.

Confidence Interval Equations

Population Mean and Proportion

Topic	Population Mean	Population Proportion
Best point estimate	$\bar{x}$	$\hat{p}$
Critical value	$z_{\alpha/2}$ or $t_{\alpha/2}$	$z_{\alpha/2}$
Margin of error	$E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ or $E = t_{\alpha/2} \frac{s}{\sqrt{n}}$	$E = z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Confidence interval	$\bar{x} \pm E$	$\hat{p} \pm E$

Example: For a sample mean $\bar{x} = 50$, $s = 10$, $n = 25$, $t_{0.025,24} = 2.064$, the 95% CI is $50 \pm 2.064 \times (10/5) = 50 \pm 4.128$.

Recommended Sample Size

Topic	Population Mean	Population Proportion
General equation	$n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^2$	$n = \frac{z_{\alpha/2}^2 \hat{p}(1-\hat{p})}{E^2}$

Note: Always round up to the next whole number.

Test Statistics and p-values for Hypothesis Testing

Parameter	Test Statistic
Proportion, $p$	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
Mean, $\mu$	$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$
Variance, $\sigma^2$	$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$

p-value: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.

Correlation Coefficients

Coefficient	Formula
Pearson's $r$	$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$
Spearman's $r_s$	$r_s = 1 - \frac{6 \sum d_i^2}{n(n^2-1)}$

Interpretation: Values close to 1 or -1 indicate strong linear relationships; values near 0 indicate weak or no linear relationship.

Additional Key Formulas

• Mean: $\bar{x} = \frac{\sum x_i}{n}$

• Sample standard deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

• Binomial probability: $P(x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}$