Statistics formulas

All 22 formulas in this topic

Arithmetic Meanxˉ=1n∑i=1nxi\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_iArithmetic Mean is the average value found by adding all numbers in a dataset and dividing by how many numbers there are.Medianx~=middle value of the ordered data\tilde{x} = \text{middle value of the ordered data}The median is the middle value of ordered data, most useful when your data contains outliers that would skew the mean value.Modemode=most frequent value\text{mode} = \text{most frequent value}Mode is the value that appears most frequently in a dataset; use it to identify the most common outcome when comparing preferences or observations.Range of a Data SetR=xmax⁡−xmin⁡R = x_{\max} - x_{\min}Range measures the spread of a data set by finding the difference between the maximum and minimum values, telling you how dispersed your data is.Weighted Meanxˉw=∑wixi∑wi\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}The weighted mean adjusts an average by giving different importance to each value, used when some data points matter more than others.Population Varianceσ2=1N∑(xi−μ)2\sigma^2 = \frac{1}{N}\sum (x_i - \mu)^2Population variance quantifies how spread data points are around the mean across an entire population, used instead of sample variance when you have all data.Sample Variances2=1n−1∑(xi−xˉ)2s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2Sample variance measures how spread out data points are around the mean, using n-1 instead of n to account for sample uncertainty.Standard Deviations=1n−1∑(xi−xˉ)2s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}Standard deviation measures how spread out data points are from the average—use it to understand variability in a dataset.Z-Scorez=x−μσz = \frac{x - \mu}{\sigma}The Z-Score tells you how many standard deviations a data point is from the mean, revealing whether it is typical or unusual.Interquartile RangeIQR=Q3−Q1\text{IQR} = Q_3 - Q_1The interquartile range measures the spread of the middle 50% of your data, telling you how compressed or spread out typical values are.Outlier Rulex<Q1−1.5 IQRorx>Q3+1.5 IQRx < Q_1 - 1.5\,\text{IQR} \quad \text{or} \quad x > Q_3 + 1.5\,\text{IQR}The Outlier Rule identifies data values farther than 1.5 times the interquartile range from the quartiles—use it to flag unusual observations.Empirical Rule68% within 1σ,;95% within 2σ,;99.7% within 3σ68\% \text{ within } 1\sigma, ; 95\% \text{ within } 2\sigma, ; 99.7\% \text{ within } 3\sigmaThe Empirical Rule describes where approximately 68%, 95%, and 99.7% of normally distributed data fall, measured in standard deviations from the mean.Normal Distributionf(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}The normal distribution describes the probability density of a continuous variable, used to model measurements and natural phenomena.Standard Error of the MeanSE=snSE = \frac{s}{\sqrt{n}}The standard error of the mean measures how much sample means vary around the true population mean; use it for confidence intervals and hypothesis tests.Confidence Interval for a Meanxˉ±t∗sn\bar{x} \pm t^{*}\frac{s}{\sqrt{n}}Estimates the range where a population mean likely falls using sample data when the population standard deviation is unknown.Margin of ErrorE=z∗σnE = z^{*}\frac{\sigma}{\sqrt{n}}Margin of Error measures how far your sample statistic might differ from the true population value, used when estimating populations from sample data.Correlation Coefficientr=∑(xi−xˉ)(yi−yˉ)(n−1)sxsyr = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}Correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables, ranging from -1 to 1.Least Squares Regression Liney^=b0+b1x,b1=rsysx\hat{y} = b_0 + b_1 x, \quad b_1 = r\frac{s_y}{s_x}The line of best fit that minimizes squared prediction errors, used to predict one variable from another in a linear relationship.Coefficient of DeterminationR2=proportion of variation explainedR^2 = \text{proportion of variation explained}Coefficient of Determination (R²) indicates how much of the variation in your outcome variable is explained by a regression model's predictors.Chi-Square Statisticχ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}A measure of how much observed data deviates from expected frequencies, used to test whether categorical distributions match a hypothesis.One-Sample t Testt=xˉ−μ0s/nt = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}A one-sample t test compares a sample mean to a hypothesized population mean when the population standard deviation is unknown.Five Number Summary{xmin⁡,Q1,x~,Q3,xmax⁡}\{x_{\min}, Q_1, \tilde{x}, Q_3, x_{\max}\}The Five Number Summary divides a dataset into quartiles to reveal its center, spread, and range—essential for exploratory data analysis and box plots.

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