3.1 Numerical Measures

There are differences between a population and a sample:

Measures of	Category	Population	Sample
	What is it?	Reality	A small fraction of reality (inference)
	Characteristics described by	Parameters	Statistics
Central Tendency	Mean	$\mu = E(Y)$	$\hat{\mu} = \overline{y}$
Central Tendency	Median	50th percentile	$y_{(\frac{n+1}{2})}$
Dispersion	Variance	$$\sigma^2 = var(Y) = E[(Y-\mu)^2]$$	$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (y_i - \overline{y})^2$
Dispersion	Coefficient of Variation	$\frac{\sigma}{\mu}$	$\frac{s}{\overline{y}}$
Dispersion	Interquartile Range	Difference between 25th and 75th percentiles; robust to outliers
Shape	Skewness Standardized 3rd central moment (unitless)	$g_1 = \frac{\mu_3}{\sigma^3}$	$\hat{g_1} = \frac{m_3}{m_2^{3/2}}$
Shape	Central moments	$\mu=E(Y)$, $\mu_2 = \sigma^2 = E[(Y-\mu)^2]$, $\mu_3 = E[(Y-\mu)^3]$, $\mu_4 = E[(Y-\mu)^4]$	$m_2 = \frac{1}{n} \sum_{i=1}^{n} (y_i - \overline{y})^2$ $m_3 = \frac{1}{n} \sum_{i=1}^{n} (y_i - \overline{y})^3$
Shape	Kurtosis (peakedness and tail thickness) Standardized 4th central moment	$g_2^* = \frac{E[(Y-\mu)^4]}{\sigma^4}$	$\hat{g_2} = \frac{m_4}{m_2^2} - 3$

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Notes:

1. Order Statistics: $y_{(1)}, y_{(2)}, \ldots, y_{(n)}$, where $y_{(1)} < y_{(2)} < \ldots < y_{(n)}$.

2. Coefficient of Variation:

   • Defined as the standard deviation divided by the mean.
   • A stable, unitless statistic useful for comparison.

3. Symmetry:

   • Symmetric distributions: Mean = Median; Skewness = 0.
   • Skewed Right: Mean > Median; Skewness > 0.
   • Skewed Left: Mean < Median; Skewness < 0.

4. Central Moments:

   • $\mu = E(Y)$
   • $\mu_2 = \sigma^2 = E[(Y-\mu)^2]$
   • $\mu_3 = E[(Y-\mu)^3]$
   • $\mu_4 = E[(Y-\mu)^4]$

Skewness ($\hat{g_1}$)

1. Sampling Distribution:
   For samples drawn from a normal population:
   • $\hat{g_1}$ is approximately distributed as $N(0, \frac{6}{n})$ when $n > 150$.
2. Inference:
   • Large Samples: Inference on skewness can be based on the standard normal distribution.
     The 95% confidence interval for $g_1$ is given by: $$\hat{g_1} \pm 1.96 \sqrt{\frac{6}{n}}$$
   • Small Samples: For small samples, consult special tables such as:
     • Snedecor and Cochran (1989), Table A 19(i)
     • Monte Carlo test results

Kurtosis ($\hat{g_2}$)

1. Definitions and Relationships:
   • A normal distribution has kurtosis $g_2^* = 3$.
     Kurtosis is often redefined as: $$g_2 = \frac{E[(Y - \mu)^4]}{\sigma^4} - 3$$ where the 4th central moment is estimated by: $$m_4 = \frac{\sum_{i=1}^n (y_i - \overline{y})^4}{n}$$
2. Sampling Distribution:
   For large samples ($n > 1000$):
   • $\hat{g_2}$ is approximately distributed as $N(0, \frac{24}{n})$.
3. Inference:
   • Large Samples: Inference for kurtosis can use standard normal tables.
   • Small Samples: Refer to specialized tables such as:
     • Snedecor and Cochran (1989), Table A 19(ii)
     • Geary (1936)

Kurtosis Value	Tail Behavior	Comparison to Normal Distribution
$g_2 > 0$ (Leptokurtic)	Heavier Tails	Examples: $t$-distributions
$g_2 < 0$ (Platykurtic)	Lighter Tails	Examples: Uniform or certain bounded distributions
$g_2 = 0$ (Mesokurtic)	Normal Tails	Exactly matches the normal distribution

# Generate random data from a normal distribution
data <- rnorm(100)

# Load the e1071 package for skewness and kurtosis functions
library(e1071)

# Calculate skewness
skewness_value <- skewness(data)
cat("Skewness:", skewness_value, "\n")
#> Skewness: 0.362615

# Calculate kurtosis
kurtosis_value <- kurtosis(data)
cat("Kurtosis:", kurtosis_value, "\n")
#> Kurtosis: -0.3066409

References

Geary, R Cf. 1936. “Moments of the Ratio of the Mean Deviation to the Standard Deviation for Normal Samples.” Biometrika, 295–307.

Snedecor, George W., and William G. Cochran. 1989. “Statistical Methods.”