Formula Sheet: Descriptive Statistics

Measures of Central Tendency

Mean (Arithmetic Average)

Sample Mean: \[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i = \frac{x_1 + x_2 + ... + x_n}{n} \]

• \(\bar{x}\) = sample mean
• \(x_i\) = individual data values
• \(n\) = number of observations in the sample

Population Mean: \[ \mu = \frac{1}{N}\sum_{i=1}^{N}x_i \]

• \(\mu\) = population mean
• \(N\) = total number of observations in the population

Weighted Mean: \[ \bar{x}_w = \frac{\sum_{i=1}^{n}w_i x_i}{\sum_{i=1}^{n}w_i} \]

• \(\bar{x}_w\) = weighted mean
• \(w_i\) = weight assigned to observation \(x_i\)
• \(x_i\) = individual data values
• \(n\) = number of observations

Median

Definition: The middle value when data is arranged in ascending or descending order. For odd number of observations: \[ \text{Median} = x_{\frac{n+1}{2}} \] For even number of observations: \[ \text{Median} = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2} \]

• \(n\) = number of observations
• \(x_i\) = data values arranged in order

Mode

Definition: The value that occurs most frequently in a dataset.

• A dataset can have no mode, one mode (unimodal), two modes (bimodal), or multiple modes (multimodal)
• Mode is the only measure of central tendency applicable to nominal data

Measures of Dispersion (Variability)

Range

\[ \text{Range} = x_{max} - x_{min} \]

• \(x_{max}\) = maximum value in dataset
• \(x_{min}\) = minimum value in dataset

Variance

Sample Variance: \[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 \]

• \(s^2\) = sample variance
• \(x_i\) = individual data values
• \(\bar{x}\) = sample mean
• \(n\) = number of observations in the sample
• Note: Division by \(n-1\) provides an unbiased estimate (Bessel's correction)

Population Variance: \[ \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2 \]

• \(\sigma^2\) = population variance
• \(\mu\) = population mean
• \(N\) = total number of observations in the population

Computational Formula for Sample Variance: \[ s^2 = \frac{\sum_{i=1}^{n}x_i^2 - \frac{(\sum_{i=1}^{n}x_i)^2}{n}}{n-1} \]

Standard Deviation

Sample Standard Deviation: \[ s = \sqrt{s^2} = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2} \]

• \(s\) = sample standard deviation
• Units are the same as the original data

Population Standard Deviation: \[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2} \]

• \(\sigma\) = population standard deviation

Coefficient of Variation

\[ CV = \frac{s}{\bar{x}} \times 100\% \] or for population: \[ CV = \frac{\sigma}{\mu} \times 100\% \]

• CV = coefficient of variation (expressed as percentage)
• \(s\) = sample standard deviation
• \(\bar{x}\) = sample mean
• Note: Dimensionless measure of relative variability; useful for comparing variability between datasets with different units or means

Interquartile Range (IQR)

\[ IQR = Q_3 - Q_1 \]

• IQR = interquartile range
• \(Q_3\) = third quartile (75th percentile)
• \(Q_1\) = first quartile (25th percentile)
• Measures the spread of the middle 50% of the data
• Resistant to outliers

Measures of Position

Percentiles

Position of kth Percentile: \[ L_k = \frac{k}{100}(n+1) \]

• \(L_k\) = position of the kth percentile
• \(k\) = desired percentile (0 to 100)
• \(n\) = number of observations
• Note: If \(L_k\) is not an integer, interpolate between the two nearest data values

Quartiles

• \(Q_1\) = first quartile = 25th percentile
• \(Q_2\) = second quartile = 50th percentile = median
• \(Q_3\) = third quartile = 75th percentile

Standard Score (Z-Score)

Sample Z-Score: \[ z = \frac{x - \bar{x}}{s} \] Population Z-Score: \[ z = \frac{x - \mu}{\sigma} \]

• \(z\) = standardized score
• \(x\) = data value
• \(\bar{x}\) or \(\mu\) = mean
• \(s\) or \(\sigma\) = standard deviation
• Z-score indicates how many standard deviations a value is from the mean
• Positive z-score: value is above the mean
• Negative z-score: value is below the mean

Measures of Shape

Skewness

Sample Skewness (Pearson's moment coefficient): \[ g_1 = \frac{n}{(n-1)(n-2)}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^3 \] Approximate Skewness: \[ \text{Skewness} \approx \frac{3(\bar{x} - \text{Median})}{s} \]

• \(g_1\) = sample skewness coefficient
• \(\bar{x}\) = sample mean
• \(s\) = sample standard deviation
• Interpretation:
  • \(g_1 = 0\): symmetric distribution
  • \(g_1 > 0\): positively skewed (right-skewed, tail extends to the right)
  • \(g_1 < 0\):="" negatively="" skewed="" (left-skewed,="" tail="" extends="" to="" the="">

Kurtosis

Sample Kurtosis (excess kurtosis): \[ g_2 = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)} \]

• \(g_2\) = excess kurtosis coefficient
• Interpretation:
  • \(g_2 = 0\): mesokurtic (normal distribution)
  • \(g_2 > 0\): leptokurtic (heavier tails, more peaked)
  • \(g_2 < 0\):="" platykurtic="" (lighter="" tails,="">

Correlation and Covariance

Covariance

Sample Covariance: \[ s_{xy} = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}) \] Population Covariance: \[ \sigma_{xy} = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu_x)(y_i - \mu_y) \]

• \(s_{xy}\) = sample covariance between variables x and y
• \(\bar{x}\), \(\bar{y}\) = sample means of x and y
• \(n\) = number of paired observations
• Positive covariance indicates variables tend to move together
• Negative covariance indicates variables tend to move in opposite directions

Correlation Coefficient

Pearson Correlation Coefficient (r): \[ r = \frac{s_{xy}}{s_x s_y} = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}} \] Alternative Computational Formula: \[ r = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n\sum x_i^2 - (\sum x_i)^2}\sqrt{n\sum y_i^2 - (\sum y_i)^2}} \]

• \(r\) = Pearson correlation coefficient
• \(s_x\), \(s_y\) = sample standard deviations of x and y
• Range: \(-1 \leq r \leq +1\)
• Interpretation:
  • \(r = +1\): perfect positive linear relationship
  • \(r = -1\): perfect negative linear relationship
  • \(r = 0\): no linear relationship
  • \(|r| > 0.7\): strong correlation
  • \(0.3 < |r|="">< 0.7\):="" moderate="">
  • \(|r| < 0.3\):="" weak="">

Coefficient of Determination

\[ r^2 = \text{(Pearson correlation coefficient)}^2 \]

• \(r^2\) = coefficient of determination
• Range: \(0 \leq r^2 \leq 1\)
• Represents the proportion of variance in one variable that is predictable from the other variable
• Expressed as a percentage when multiplied by 100

Linear Regression

Simple Linear Regression Model

\[ y = a + bx \] or \[ \hat{y} = a + bx \]

• \(\hat{y}\) = predicted value of dependent variable
• \(x\) = independent variable
• \(a\) = y-intercept
• \(b\) = slope of the regression line

Slope of Regression Line

\[ b = \frac{s_{xy}}{s_x^2} = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} \] Alternative Computational Formula: \[ b = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2} \]

• \(b\) = slope (regression coefficient)
• \(s_{xy}\) = covariance of x and y
• \(s_x^2\) = variance of x

Y-Intercept of Regression Line

\[ a = \bar{y} - b\bar{x} \]

• \(a\) = y-intercept
• \(\bar{x}\), \(\bar{y}\) = means of x and y
• \(b\) = slope
• The regression line always passes through the point \((\bar{x}, \bar{y})\)

Residual

\[ e_i = y_i - \hat{y}_i \]

• \(e_i\) = residual for observation i
• \(y_i\) = actual observed value
• \(\hat{y}_i\) = predicted value from regression equation

Sum of Squared Errors (SSE)

\[ SSE = \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 = \sum_{i=1}^{n}e_i^2 \]

• SSE = sum of squared errors (residuals)
• Measure of variation not explained by the regression model

Standard Error of Estimate

\[ s_e = \sqrt{\frac{SSE}{n-2}} = \sqrt{\frac{\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}{n-2}} \]

• \(s_e\) = standard error of the estimate
• \(n\) = number of observations
• Measures the typical deviation of observed values from the regression line
• Smaller values indicate better fit

Frequency Distributions

Class Width

\[ \text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}} = \frac{x_{max} - x_{min}}{k} \]

• \(k\) = number of classes
• Typically round up to a convenient number

Sturges' Rule for Number of Classes

\[ k = 1 + 3.322 \log_{10}(n) \] or \[ k = 1 + \frac{\log(n)}{\log(2)} \]

• \(k\) = suggested number of classes
• \(n\) = number of observations
• Provides a starting point; final number may be adjusted

Class Midpoint

\[ \text{Midpoint} = \frac{\text{Lower Class Limit} + \text{Upper Class Limit}}{2} \]

Relative Frequency

\[ \text{Relative Frequency} = \frac{\text{Class Frequency}}{n} \]

• \(n\) = total number of observations
• Sum of all relative frequencies equals 1.0

Cumulative Frequency

• Sum of frequencies up to and including the current class
• The cumulative frequency of the last class equals \(n\)

Outlier Detection

Interquartile Range (IQR) Method

Lower Fence: \[ \text{Lower Fence} = Q_1 - 1.5 \times IQR \] Upper Fence: \[ \text{Upper Fence} = Q_3 + 1.5 \times IQR \]

• Data points below the lower fence or above the upper fence are considered outliers
• \(Q_1\) = first quartile
• \(Q_3\) = third quartile
• \(IQR\) = interquartile range = \(Q_3 - Q_1\)

Z-Score Method

• A data point is typically considered an outlier if:
• \(|z| > 2\) (for small samples)
• \(|z| > 3\) (for large samples or more conservative approach)
• Where \(z\) is the z-score of the data point

Special Statistical Properties

Properties of Mean

• Sum of deviations from the mean equals zero: \(\sum_{i=1}^{n}(x_i - \bar{x}) = 0\)
• Mean is sensitive to outliers and extreme values
• Mean of a constant times a variable: \(\overline{kx} = k\bar{x}\)
• Mean of sum/difference: \(\overline{x \pm y} = \bar{x} \pm \bar{y}\)

Properties of Variance

• Variance of a constant is zero: \(\text{Var}(k) = 0\)
• Variance of a constant times a variable: \(\text{Var}(kx) = k^2 \text{Var}(x)\)
• For independent variables: \(\text{Var}(x \pm y) = \text{Var}(x) + \text{Var}(y)\)

Properties of Standard Deviation

• Standard deviation of a constant is zero: \(\text{SD}(k) = 0\)
• Standard deviation of a constant times a variable: \(\text{SD}(kx) = |k| \times \text{SD}(x)\)

Empirical Rule (68-95-99.7 Rule)

For normal (bell-shaped) distributions:

• Approximately 68% of data falls within \(\mu \pm \sigma\) (one standard deviation from mean)
• Approximately 95% of data falls within \(\mu \pm 2\sigma\) (two standard deviations from mean)
• Approximately 99.7% of data falls within \(\mu \pm 3\sigma\) (three standard deviations from mean)

Chebyshev's Theorem

For any distribution (regardless of shape): \[ \text{Minimum proportion} = 1 - \frac{1}{k^2} \]

• \(k\) = number of standard deviations from the mean (\(k > 1\))
• At least \(1 - \frac{1}{k^2}\) of the data falls within \(k\) standard deviations of the mean
• Examples:
  • \(k = 2\): at least 75% of data within \(\mu \pm 2\sigma\)
  • \(k = 3\): at least 89% of data within \(\mu \pm 3\sigma\)