Descriptive Statistics

For Ungrouped Data

Measures of Central Tendency

Arithmetic Mean

$$\overline{X} = \frac{1}{n} \sum_{i=1}^{n} x_i$$

Median

$$\tilde{X} = \begin{cases} x_{\left(\frac{n+1}{2}\right)} & \text{if } n \text{ is odd} \\[10pt] \dfrac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2} & \text{if } n \text{ is even} \end{cases}$$

Mode

$$Mo = \text{value(s) with the highest absolute frequency}$$

Measures of Dispersion

Range

$$R = x_{\max} - x_{\min}$$

Variance (population)

$$S^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{X})^2$$

Practical formula:

$$S^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \overline{X}^2$$

Standard Deviation

$$S = \sqrt{S^2}$$

Coefficient of Variation

$$CV = \frac{S}{\overline{X}}$$

Measures of Shape

Skewness Coefficient

$$\alpha_3 = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{X})^3}{S^3}$$

• $\alpha_3 = 0$: Symmetric distribution
• $\alpha_3 > 0$: Positive skew (tail to the right)
• $\alpha_3 < 0$: Negative skew (tail to the left)

Kurtosis Coefficient

$$\alpha_4 = \frac{\frac{1}{n} \sum_{i=1}^{n} (x_i - \overline{X})^4}{S^4}$$

• $\alpha_4 = 3$: Mesokurtic distribution (like the normal)
• $\alpha_4 > 3$: Leptokurtic (more peaked)
• $\alpha_4 < 3$: Platykurtic (less peaked)

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For Grouped Data (Frequency Tables)

Measures of Central Tendency

Arithmetic Mean

$$\overline{X} = \frac{1}{n} \sum_{i=1}^{m} f_i \cdot x_i$$

where $x_i$ is the class mark (midpoint).

Median

$$\tilde{X} = L_{\text{med}} + \left[ \frac{\frac{n}{2} - F_{i-1}}{f_{\text{med}}} \right] \cdot c_{\text{med}}$$

• $L_{\text{med}}$: Lower limit of the median class
• $f_{\text{med}}$: Frequency of the median class
• $F_{i-1}$: Cumulative frequency preceding the median class
• $c_{\text{med}}$: Width of the median class

Mode

$$Mo = L_{\text{Mo}} + \left[ \frac{\Delta_1}{\Delta_1 + \Delta_2} \right] \cdot c_{\text{Mo}}$$

• $\Delta_1 = f_{\text{Mo}} - f_{\text{Mo}-1}$
• $\Delta_2 = f_{\text{Mo}} - f_{\text{Mo}+1}$
• $L_{\text{Mo}}$: Lower limit of the modal class

Measures of Dispersion

Range

$$R = L_{\text{sup, last}} - L_{\text{inf, first}}$$

Variance

$$S^2 = \frac{1}{n} \sum_{i=1}^{m} f_i (x_i - \overline{X})^2 = \frac{\sum_{i=1}^{m} f_i x_i^2}{n} - \overline{X}^2$$

Standard Deviation

$$S = \sqrt{S^2}$$

Coefficient of Variation

$$CV = \frac{S}{\overline{X}}$$

Measures of Shape

Skewness Coefficient

$$\alpha_3 = \frac{\frac{1}{n} \sum_{i=1}^{m} f_i (x_i - \overline{X})^3}{S^3}$$

Kurtosis Coefficient

$$\alpha_4 = \frac{\frac{1}{n} \sum_{i=1}^{m} f_i (x_i - \overline{X})^4}{S^4}$$

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Fractiles (Quantiles) for Grouped Data

General Formula

$$P_k = L_k + \left[ \frac{n \cdot k - F_{i-1}}{f_k} \right] \cdot c_k$$

where:

• $P_k$: Desired fractile
• $L_k$: Lower limit of the fractile class
• $k$: Corresponding proportion (e.g., 0.25 for $Q_1$)
• $f_k$: Frequency of the fractile class
• $F_{i-1}$: Preceding cumulative frequency
• $c_k$: Width of the fractile class

Types of Fractiles

Type	Symbol	Proportion (k)	Example
Quartiles	$Q_1, Q_2, Q_3$	0.25, 0.50, 0.75	$Q_3$: $k = 0.75$
Deciles	$D_1, D_2, \ldots, D_9$	0.10, 0.20, $\ldots$, 0.90	$D_5$ = Median
Percentiles	$P_1, P_2, \ldots, P_{99}$	0.01, 0.02, $\ldots$, 0.99	$P_{90}$: $k = 0.90$

Important relationships:

• $Q_2 = D_5 = P_{50}$ = Median
• $D_1 = P_{10}$, $D_9 = P_{90}$

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Practical Considerations

1. Grouped data: All formulas use the class mark (midpoint) $x_i$ as the representative value of the interval.

2. Median and fractiles: Their calculation first requires identifying the corresponding class by analyzing cumulative frequencies.

3. Mode: A distribution can be unimodal, bimodal, or multimodal. For grouped data, interpolation within the modal class is used.

4. Interpreting the coefficient of variation:

   • $CV < 15\%$: Low relative dispersion
   • $15\% \leq CV \leq 30\%$: Moderate dispersion
   • $CV > 30\%$: High relative dispersion

5. Units of measurement:

   • Variance retains the original units squared
   • Standard deviation maintains the original units
   • The coefficients of variation, skewness, and kurtosis are dimensionless