Describing Data and Statistical Measures in Econometrics

The shape of a distribution is a critical descriptor:

• Symmetric: The left half is a mirror image of the right half.
• Right-skewed (Positively skewed): A long tail stretches to the right. This is common in economics for variables like income, wealth, and house prices.
• Left-skewed (Negatively skewed): A long tail stretches to the left.
• Bimodal: The distribution has two distinct peaks.

Measures of Central Tendency

How do we find the "center" of a dataset? We typically use one of three candidates:

1. Sample Mean ($\bar{X}$): The arithmetic average, calculated using the summation operator ($\sum$): $\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i$ While useful, the mean is sensitive to extreme values (outliers).
2. Sample Median: The middle value when the data is sorted from smallest to largest. If $n$ is odd, it is the middle observation; if $n$ is even, it is the average of the two middle observations.
3. Sample Mode: The value that occurs most frequently.

Exam Tip: Resistant Statistics A statistic is "resistant" (or robust) if a small number of extreme observations barely moves it. The median is a resistant statistic, whereas the mean is not. In a right-skewed distribution like household income, the mean is typically much larger than the median.

Measures of Variability

Knowing the center is only half the story; we also need to know how spread out the data is.

• Range: The simplest measure, calculated as $X_{max} - X_{min}$.
• Sample Variance ($s^2_X$): The average squared deviation from the mean: $s^2_X = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2$
• Sample Standard Deviation ($s_X$): The square root of the variance, which returns the measure to the original units of the data: $s_X = \sqrt{s^2_X}$
• Percentiles and Quartiles: The $p$-th percentile is the value below which $p%$ of the observations fall. The quartiles are specific percentiles: $Q_1$ (25th), $Q_2$ (50th/Median), and $Q_3$ (75th).
• Interquartile Range (IQR): The distance between the third and first quartiles ($Q_3 - Q_1$), often used to describe the spread of the middle 50% of the data.

Identifying Outliers

An outlier is an observation far away from the bulk of the data. A common rule of thumb is to flag any observation $X_i$ if: $X_i < Q_1 - 1.5 \cdot \text{IQR} \quad \text{or} \quad X_i > Q_3 + 1.5 \cdot \text{IQR}$

Relationships Between Variables

Economics is often concerned with how two variables relate, such as how education affects wages.

Scatterplots and Covariance

A scatterplot is a graphical representation of paired observations $(X_i, Y_i)$, where each dot represents one observation. To quantify this relationship, we use sample covariance ($s_{XY}$), which measures how two variables move together: $s_{XY} = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})$

Correlation and Elasticity

Because the magnitude of covariance is hard to interpret, we use the sample correlation coefficient ($r_{XY}$), which is unit-free and ranges from $-1$ to $+1$: $r_{XY} = \frac{s_{XY}}{s_X s_Y}$

Important Warning: Correlation is not Causation Just because two variables are correlated does not mean one causes the other. A correlation might appear due to an omitted factor, reverse causality, or selection bias.

Finally, economists often use elasticity ($\epsilon_{yx}$) to describe the sensitivity of one variable to another. It is defined as the percentage change in $y$ per percentage change in $x$. Using the natural logarithm ($\log$), we can approximate elasticity for small changes: $\epsilon_{yx} \approx \frac{\Delta \log(y)}{\Delta \log(x)}$