Descriptive Statistics: Numerical Measures

Measures of Location

  • Parameter versus Statistic:

    • Parameter: A measurable characteristic of a population.

    • Statistic: A measurable characteristic of a sample.

  • Arithmetic Mean:

    • Requires interval or ratio level scale.

    • Uses all data values and is unique.

    • The sum of deviations from the mean is zero: ∑(X−Xˉ)=0\sum (X - \bar{X}) = 0.

    • Population Mean: μ=∑XN\mu = \frac{\sum X}{N}

    • Sample Mean: Xˉ=∑Xn\bar{X} = \frac{\sum X}{n}

  • Weighted Mean:

    • Accounts for values with varying importance or frequencies.

    • Formula: Xˉw=w1X1+w2X2+⋯+wnXnw1+w2+⋯+wn\bar{X}_w = \frac{w_1 X_1 + w_2 X_2 + \dots + w_n X_n}{w_1 + w_2 + \dots + w_n}

  • Median:

    • Midpoint of values ordered from smallest to largest.

    • Unique for each data set and unaffected by extreme values.

  • Mode:

    • Value appearing most frequently.

    • Can be identified for nominal, ordinal, interval, and ratio data.

Distribution Shape and Geometric Mean

  • Skewness and Relative Positions:

    • Symmetric (Zero Skewness): Mode=Median=Mean\text{Mode} = \text{Median} = \text{Mean}

    • Positive Skewness (Right-Skewed): Mode<Median<Mean\text{Mode} < \text{Median} < \text{Mean}

    • Negative Skewness (Left-Skewed): Mode>Median>Mean\text{Mode} > \text{Median} > \text{Mean}

Relationships between mean, median, and mode in symmetric and skewed distributions

  • Geometric Mean:

    • Measures average change of percentages, ratios, or growth rates over time.

    • Always less than or equal to the arithmetic mean.

    • Formula: GM=X1⋅X2…XnnGM = \sqrt[n]{X_1 \cdot X_2 \dots X_n}

    • Average rate of growth over time: GM=Value at end of periodValue at beginning of periodn−1GM = \sqrt[n]{\frac{\text{Value at end of period}}{\text{Value at beginning of period}}} - 1

Measures of Dispersion

  • Range:

    • Difference between highest and lowest values: Range=Largest value−Smallest value\text{Range} = \text{Largest value} - \text{Smallest value}

  • Mean Deviation:

    • Arithmetic mean of absolute deviations from the mean: MD=∑∣X−Xˉ∣nMD = \frac{\sum |X - \bar{X}|}{n}

  • Variance and Standard Deviation:

    • Population Variance: σ2=∑(X−μ)2N\sigma^2 = \frac{\sum (X - \mu)^2}{N}

    • Population Standard Deviation: σ=∑(X−μ)2N\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}

    • Sample Variance: s2=∑(X−Xˉ)2n−1s^2 = \frac{\sum (X - \bar{X})^2}{n - 1}

    • Sample Standard Deviation: s=∑(X−Xˉ)2n−1s = \sqrt{\frac{\sum (X - \bar{X})^2}{n - 1}}

Dispersion Rules and Grouped Data

  • Chebyshev's Theorem:

    • For any constant k>1k > 1, the proportion of values lying within kk standard deviations of the mean is at least 1−1k21 - \frac{1}{k^2}.

  • Empirical Rule:

    • Applies to symmetrical, bell-shaped distributions:

    • ≈68%\approx 68\% of observations lie within μ±1σ\mu \pm 1\sigma

    • ≈95%\approx 95\% of observations lie within μ±2σ\mu \pm 2\sigma

    • ≈99.7%\approx 99.7\% of observations lie within μ±3σ\mu \pm 3\sigma

Empirical Rule curve showing 68%, 95%, and 99.7% distribution

  • Grouped Data Calculations:

    • Arithmetic Mean: Xˉ=∑fMn\bar{X} = \frac{\sum fM}{n} (where MM is class midpoint, ff is class frequency)

    • Sample Standard Deviation: s=∑f(M−Xˉ)2n−1s = \sqrt{\frac{\sum f(M - \bar{X})^2}{n - 1}}