Topic 1. Descriptive Statistics

Flashcards for studying position, dispersion, and form statistics from Topic 1: Descriptive Statistics.

Role of central tendency statistics

Statistics that determine values around which a distribution is grouped. Categories include the Mean (Arithmetic, Geometric, Harmonic), Median, and Mode.

Formula for Arithmetic Mean ($\bar{x}$)

$$\bar{x} = \frac{\sum x_i \cdot n_i}{N}$$

Formula for Geometric Mean ($G$)

$$G = \sqrt[N]{x_1^{n_1} \cdot x_2^{n_2} \cdots x_n^{n_n}}$$

Formula for Harmonic Mean ($H$)

$$H = \frac{N}{\frac{1}{x_1} \cdot n_1 + \frac{1}{x_2} \cdot n_2 + \dots + \frac{1}{x_n} \cdot n_n}$$

Definition and common types of position quantiles

Position statistics that divide a distribution into parts with the same number of observations. Common types include Quartiles, Deciles, and Percentiles.

General formula for Quantiles ($Q_{r/k}$)

$$Q_{r/k} = L_{i-1} + \frac{\frac{r}{k} \cdot N - N_{i-1}}{n_i} \cdot c_i$$

Meaning of variables $r$ and $k$ in the quantile formula

$r$ is the order of the quantile, and $k$ is the total number of equal intervals into which the distribution is divided.

The five sample dispersion statistics

1. Range ($R_a$)

2. Interquartile range ($R_I$)

3. Variance ($S^2$)

4. Standard deviation ($S$)

5. Coefficient of variation ($CV$)

Formulas for Range ($R_a$) and Interquartile Range ($R_I$)

Range: $R_a = x_n - x_1$

Interquartile range: $R_I = Q_3 - Q_1$

Formula for sample Variance ($S^2$)

$$S^2 = \frac{\sum_i x_i^2 \cdot n_i}{N} - \bar{x}^2$$

Formula for Standard Deviation ($S$)

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

Formula for Coefficient of Variation ($CV$)

$$CV = \frac{S}{\bar{x}}$$

Representativeness of the mean when $0 \le CV \le 0.3$

The mean ($\bar{x}$) is very representative.

Representativeness of the mean when $0.3 < CV \le 0.5$

The mean ($\bar{x}$) has medium-high representativeness.

Representativeness of the mean when $0.5 < CV \le 0.7$

The mean ($\bar{x}$) has medium-low representativeness.

Representativeness of the mean when $CV > 0.7$

The mean ($\bar{x}$) has low representativeness.

What sample form statistics for symmetry measure

They measure the symmetry of the distribution around its mean, most commonly using the Fisher Asymmetry Coefficient ($g_1$).

Formula for Fisher Asymmetry Coefficient ($g_1$)

$$g_1 = \frac{\frac{\sum (x_i - \bar{x})^3 \cdot n_i}{N}}{S^3}$$

Classification of distribution symmetry based on Fisher Asymmetry Coefficient ($g_1$)

• $g_1 = 0$: Symmetric distribution

• $g_1 > 0$: Right asymmetric or positive distribution

• $g_1 < 0$: Left asymmetric or negative distribution

What pointing (kurtosis) measures

It measures the pointing of the frequency polygon compared to a Normal distribution, most commonly using the Pointing Coefficient or Kurtosis ($g_2$).

Formula for Pointing Coefficient or Kurtosis ($g_2$)

$$g_2 = \frac{\frac{\sum (x_i - \bar{x})^4 \cdot n_i}{N}}{S^4} - 3$$

Classification of pointing based on Pointing Coefficient / Kurtosis ($g_2$)

• $g_2 > 0$: More pointed than Normal

• $g_2 = 0$: Same pointed as Normal

• $g_2 < 0$: Less pointed than Normal