QMB3200- Averages, Spread, Shape, Z-scores, and Outliers

Mean (average)

Add up all the observations and divide by the number of observations. Sample mean: x̄. Population mean: μ.

Notation

Σ means “add up”. xi is the ith piece of data (in 1, 2, 3, 4, 5, x4 = 4). n is the number of observations in a sample. N is the number in a population.

Median

The middle value when the data are in ascending order. Odd number of observations: the middle value. Even number: the average of the two middle values. It is less affected by extreme values than the mean

Mode

The value that occurs most often. Two modes: bimodal. Three or more modes: multimodal.

Weighted mean

Used when observations have different weights (relative importance): Σ(wixi) / Σwi. Example: classwork 30%, exam 60%, attendance 10%.

Geometric mean

The nth root of the product of n values: (x1·x2·...·xn)1/n. Used for growth rates. Values must be positive.

Percentile

Shows how the data are spread from the smallest to the largest value. About p% of the values are at or below the pth percentile and about (100 − p)% are at or above it. The 50th percentile is the median.

Quartiles

Show how the data are spread over four parts of about 25% each. Q1 = 25th percentile, Q2 = 50th percentile (the median), Q3 = 75th percentile

Range

Largest value − smallest value.

Interquartile range (IQR)

Q3 − Q1. The range of the middle 50% of the data.

Variance

Measures spread using all the data, based on how far each value is from the mean (its deviation). Sample variance: s2. Population variance: σ2. Useful for comparing the variability of two or more variables, but its units are squared.

Standard deviation

The square root of the variance. Its units are not squared, so it is easier to compare with the mean and other statistics. Variance = (standard deviation)².

Sample formulas

s² = Σ(xi − x̄)² / (n − 1) s = √s²

Population formulas

σ² = Σ(xi − μ)² / N σ = √σ²

Coefficient of variation

How large the standard deviation is relative to the mean.

Skewness

Measures the shape of a distribution. Skewed to the left: negative. Skewed to the right: positive. Symmetric: zero.

Mean vs. median

Symmetric: the mean and median are equal. Positively skewed: the mean is usually greater than the median. Negatively skewed: the mean is usually less than the median.

z-score

The number of standard deviations a value is from the mean. It measures the relative location of a value in the dataset.
Sample: z = (x − x̄) / s. Population: z = (x − μ) / σ.

Chebyshev’s theorem

Works for any dataset. At least 1 − 1/z² of the values are within z standard deviations of the mean (z > 1). For z = 2: at least 75%. For z = 3: at least 88.9%.

Empirical rule

For bell-shaped data: about 68% of values are within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3.

Outliers

Extreme values compared with the rest of the data. z-score rule of thumb: |z| > 3 (below −3 or above +3). IQR rule: below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).