OddClasses — Odd ones figure it out.

Measures of Central Tendency & Spread

Mean:

\bar{x} = \dfrac{\sum x}{n}

| Median: middle value when data is sorted (for even n, average the two middle values) | Mode: the value that occurs most frequently

Theorem 1📌 Learn for exam

Quartiles and IQR

Sort the data. Then:

$Q_1$ = lower quartile — median of the lower half (25th percentile)
$Q_2$ = median — middle value (50th percentile)
$Q_3$ = upper quartile — median of the upper half (75th percentile)

\text{IQR} = Q_3 - Q_1

A value is an outlier if:

x < Q_1 - 1.5 \times \text{IQR} \quad \text{or} \quad x > Q_3 + 1.5 \times \text{IQR}

Theorem 2📌 Learn for exam

Variance and Standard Deviation

\sigma^2 = \frac{\sum(x - \bar{x})^2}{n} \qquad \text{(population variance)}

\sigma = \sqrt{\sigma^2} \qquad \text{(standard deviation)}

The standard deviation measures how spread out the data is around the mean. A larger $\sigma$ means the data is more spread out; a smaller $\sigma$ means data clusters tightly around the mean.

Tip

Standard deviation is always ≥ 0. If all values in the data set are equal, then σ = 0.

Worked Example

Quartiles, IQR, and outlier test

Data: 8, 12, 15, 15, 18, 22, 25. Find Q₁, Q₃, IQR, and determine whether 25 is an outlier.

Worked Example

Standard deviation

Find σ for the data set: 4, 8, 6, 5, 7.

Statistics

Measures of Central Tendency & Spread

Quartiles, IQR, and outlier test

Standard deviation