- 95% Confidence Interval
- \(sample\;statistic \pm 2 (standard\;error)\)
- Between Groups (Numerator) Degrees of Freedom
-
\(df_{between}=k-1\)
-
\(k\) = number of groups
- Binomial Random Variable Probability
-
\(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)
-
\(n\) = number of trials
\(k\) = number of successes
\(p\) = probability event of interest occurs on any one trial
- Chi-Square (\(\chi^2\)) Test Statistic
-
\(\chi^2=\Sigma \frac{(Observed-Expected)^2}{Expected}\)
- Complement of A
- \(P(A^{C})=1−P(A)\)
- Conditional Probability of A Given B
-
\(P(A\mid B)=\frac{P(A \: \cap\: B)}{P(B)}\)
- Conditional Probability of B Given A
-
\(P(B\mid A)=\frac{P(A \: \cap\: B)}{P(A)}\)
- Confidence Interval for a Population Mean
-
\(\overline{x} \pm t^{*} \frac{s}{\sqrt{n}}\)
- Confidence Interval for the Difference Between Two Paired Means
-
\(\overline{x}_d \pm t^* \left(\frac{s_d}{\sqrt{n}}\right)\)
-
\(t^*\) is the multiplier with \(df = n-1\)
- Confidence Interval for the Difference Between Two Proportions
- \((\widehat{p}_1-\widehat{p}_2) \pm z^\ast {\sqrt{\frac{\widehat{p}_1 (1-\widehat{p}_1)}{n_1}+\frac{\widehat{p}_2 (1-\widehat{p}_2)}{n_2}}}\)
- Confidence Interval for Two Independent Means
-
\((\bar{x}_1-\bar{x}_2) \pm t^\ast{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)
- Confidence Interval of \(p\): Normal Approximation Method
-
\(\widehat{p} \pm z^{*} \left ( \sqrt{\frac{\hat{p} (1-\hat{p})}{n}} \right)\)
- Confidence Interval of \(\beta_1\)
-
\(b_1 \pm t^\ast (SE_{b_1})\)
-
\(b_1\) = sample slope
\(t^\ast\) = value from the \(t\) distribution with \(df=n-2\)
\(SE_{b_1}\) = standard error of \(b_1\)
- Degrees of Freedom: Chi-Square Test of Independence
-
\(df=(number\;of\;rows-1)(number\;of\;columns-1)\)
- Estimated Degrees of Freedom
-
\(df=smallest\;n-1\)
- Expected Cell Value
-
\(E=\frac{row\;total \; \times \; column\;total}{n}\)
- Expected Count
-
\(Expected\;count=n (p_i)\)
-
\(n\) is the total sample size
\(p_i\) is the hypothesized proportion of the "ith" group
- Finding Sample Size for Estimating a Population Proportion
-
\(n=\left ( \frac{z^*}{M} \right )^2 \tilde{p}(1-\tilde{p})\)
-
\(M\) is the margin of error
\(\tilde p\) is an estimated value of the proportion
- Finding the Sample Size for Estimating a Population Mean
-
\(n=\frac{z^{2}\widetilde{\sigma}^{2}}{M^{2}}=\left ( \frac{z\widetilde{\sigma}}{M} \right )^2\)
-
\(z\) = z multiplier for given confidence level
\(\widetilde{\sigma}\) = estimated population standard deviation
\(M\) = margin of error
- Five Number Summary
-
(Minimum, \(Q_1\), Median, \(Q_3\), Maximum)
- General Form of 95% Confidence Interval
-
\(sample\ statistic\pm2\ (standard\ error)\)
- General Form of a Test Statistic
-
\(test\;statistic=\frac{sample\;statistic-null\;parameter}{standard\;error}\)
- Interquartile Range
-
\(IQR = Q_3 - Q_1\)
- Intersection
-
\(P(A\cap B) =P(A)\times P(B\mid A)\)
- Mean of a Binomial Random Variable
-
\(\mu=np\)
Also known as \(E(X)\)
- Observed Sample Mean Difference
-
\(\overline{x}_d=\frac{\Sigma{x}_d}{n}\)
-
\(x_d\) = observed difference
- Odds
-
\(odds = \frac {number \;with \;the\; outcome}{number \;without \;the \;outcome}\)
OR
\(odds=\frac{risk}{1-risk}\)
- Pearson's r: Conceptual Formula
-
\(r=\frac{\sum{z_x z_y}}{n-1}\)
where \(z_x=\frac{x - \overline{x}}{s_x}\) and \(z_y=\frac{y - \overline{y}}{s_y}\)
- Pooled Estimate of \(p\)
-
\(\widehat{p}=\frac{\widehat{p}_1n_1+\widehat{p}_2n_2}{n_1+n_2}\)
- Population Mean
-
\(\mu=\frac{\Sigma x}{N}\)
- Power
-
\(Power = 1-\beta\)
-
\(\beta\) = probability of committing a Type II Error.
- Probability of Event A
- \(P(A)=\frac{Number\;in\;group\;A}{Total\;number}\)
- Proportion
-
\(Proportion=\frac{Number\;in\;the\;category}{Total\;number}\)
- Range
-
\(Range = Maximum - Minimum\)
- Relative Risk
- \(Relative\ Risk=\frac{Risk\ in\ Group\ 1}{Risk\ in\ Group\ 2}\)
- Residual
-
\(e_i =y_i -\widehat{y}_i\)
-
\(y_i\) = actual value of y for the ith observation
\(\widehat{y}_i\) = predicted value of y for the ith observation
- Risk
-
\(Risk= \frac{number \;with \;the\; outcome}{total\;number\;of\;outcomes}\)
- Sample Standard Deviation
-
\(s=\sqrt{\frac{\sum (x-\overline{x})^{2}}{n-1}}\)
- Sample Variance
-
\(s^{2}=\frac{\sum (x-\overline{x})^{2}}{n-1}\)
- Simple Linear Regression Line: Population
-
\(\widehat{y}=\alpha+\beta x\)
- Simple Linear Regression Line: Sample
-
\(\widehat{y}=a+bx\)
-
\(\widehat{y}\) = predicted value of \(y\) for a given value of \(x\)
\(a\) = \(y\)-intercept
\(b\) = slope
- Slope
-
\(b_1 =r \frac{s_y}{s_x}\)
-
\(r\) = Pearson’s correlation coefficient between \(x\) and \(y\)
\(s_y\) = standard deviation of \(y\)
\(s_x\) = standard deviation of \(x\)
- Standard Deviation of a Binomial Random Variable
-
\(\sigma=\sqrt {np(1-p)}\)
- Standard Deviation of the Differences
- \(s_d=\sqrt{\frac{\sum (x_d-\overline{x}_d)^{2}}{n-1}}\)
- Standard Error
-
\(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\)
- Sum of Squared Residuals
-
Also known as Sum of Squared Errors (SSE)
\(SSE=\sum (y-\widehat{y})^2\)
- Sum of Squares
-
\(SS={\sum (x-\overline{x})^{2}}\)
- Test Statistic
-
\(test\; statistic = \frac{sample \; statistic - null\;parameter}{standard \;error}\)
- Test Statistic for Dependent Means
-
\(t=\frac{\bar{x}_d-\mu_0}{\frac{s_d}{\sqrt{n}}}\)
-
\(\overline{x}_d\) = observed sample mean difference
\(\mu_0\) = mean difference specified in the null hypothesis
\(s_d\) = standard deviation of the differences
\(n\) = sample size (i.e., number of unique individuals)
- Test Statistic for Dependent Means
-
\(t=\frac{\bar{x}_d-\mu_0}{\frac{s_d}{\sqrt{n}}}\)
-
\(\overline{x}_d\) = observed sample mean difference
\(\mu_0\) = mean difference specified in the null hypothesis
\(s_d\) = standard deviation of the differences
\(n\) = sample size (i.e., number of unique individuals)
- Test Statistic for Two Independent Proportions
-
\(z=\frac{\widehat{p}_1-\widehat{p}_2}{SE_0}\)
- Test statistic: One Group Proportion
-
\(z=\frac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}\)
-
\(\widehat{p}\) = sample proportion
\(p_{0}\) = hypothesize population proportion
\(n\) = sample size
- Union
- \(P(A\cup B) = P(A)+P(B)-P(A\cap B)\)
- Within Groups (Denominator, Error) Degrees of Freedom
-
\(df_{within}=n-k\)
-
\(n\) = total sample size with all groups combined
\(k\) = number of groups
- y-intercept
-
\(b_0=\overline {y} – b_1 \overline {x}\)
-
\(\overline {y}\) = mean of \(y\)
\(\overline {x}\) = mean of \(x\)
\(b_1\) = slope
- z Test Statistic: One Group Mean
-
\(z=\frac{\overline{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}\)
-
\(\overline{x}\) = sample mean
\(\mu_{0}\) = hypothesized population mean
\(s\) = sample standard deviation
\(n\) = sample size
- z-score
-
\(z=\frac{x - \overline{x}}{s}\)
-
\(x\) = original data value
\(\overline{x}\) = mean of the original distribution
\(s\) = standard deviation of the original distribution