ECN 102: Analysis of Economics Data

Chapter 2: Univariate Data

Remy Beauregard

Summary Statistics

Overview

We will start our discussion of univariate data with summary or descriptive statistics:

Population size: \(N\) (big)
Sample size: \(n\) (little)
Measures of central tendency: where is the data centered?
Measures of spread: how squished or spread out is the data?

As before, we denote cross-sectional data \(x_i\), where \(i\) is the index of the observation. What would \(x_1\) be? \(x_{n-1}\)? \(x_n\)?

\(x_1\) is the first observation in our data, \(x_{n-1}\) is the second-to-last observation, and \(x_n\) is the last observation in our sample.

Central Tendency: Mean

Central tendency: sample mean (arithmetic mean)

\[\bar{x} = \frac{x_1+x_2+...+x_n}{n}\]

\[\bar{x} = \frac1n \sum_{i=1}^n x_i\]

Population mean

\[\mu = \frac1N\sum_{i=1}^N x_i\]

Percentiles and Quartiles

We can divide our data into quartiles, deciles, percentiles, and more. The name indicates how many pieces we cut our data into:

quartile: data is divided into 4 pieces
decile: data is divided into 10 pieces
percentile: data is divided into 100 pieces

We often talk about the upper or lower quartile, the value that contains 25% of the data below or above it. We denote these \(p^{75}\) and \(p^{25}\). We give the middle quartile (or 50^th percentile) a special name.

Central Tendency: Median

Central tendency: sample median

\(\Rightarrow\) The median is the 50th percentile of ordered data, or the value 50% of my data is above and 50% is below. Its formula depends on whether \(n\) is odd or even:

Odd: \[Median=x_{(n+1)/2}\]

Even: \[Median=\frac{x_{(n/2)+1}+x_{n/2}}{2}\]

Note that we need to order our data from small to large to apply this formula for the median.

Median vs. Mean: Outlier Robustness

The sample median will be less affected by outliers than the sample mean, as it only considers at most two central observations instead of all observations.

We use the median in cases where the data is likely to be very left- or right-skewed by outliers, such as a sample of net worth that includes someone like Elon Musk.

Example: Mean vs. Median

Histogram of simulated net worth data. The distribution is heavily right-skewed due to a large outlier near 85. A red vertical line marks the mean and a blue vertical line marks the median, illustrating that the mean lies to the right of the median in right-skewed data.

Histogram of simulated net worth data with mean and median.

Central Tendency: Mode

Central tendency: mode

The mode is the most represented value in the data. Data can be multi-modal, where multiple values are equally represented most often in the data.
We may want to invoke the mode when data is discrete or continuous but highly rounded.

Spread: Range and IQR

Spread: range and interquartile range (IQR)

Range is hopefully familiar, \(\max\{x_i\}-\min\{x_i\}\)
IQR is the difference between 75^th and 25^th percentiles, \(p^{75}-p^{25}\). The IQR captures the middle 50% of our data,
also useful if we are concerned about outliers.

Spread: Sample Variance and Std Dev

Spread: sample variance

Sample variance is the sum of squared deviations in our data, \((x_i-\bar{x})^2\). We cannot simply use \(x_i-\bar{x}\), as this will always sum to zero. We denote this: \[s^2=\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2\]
To return our measure of spread to the units of our original data, we take the square root and obtain the sample standard deviation: \[s=\sqrt{s^2} = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2}\]

Spread: Population Variance and Std Dev

Spread: population variance, std dev

\(\Rightarrow\) Like before, we can also compute population parameters for spread if we know our entire population. However, this will require a slight change to the formula, which we will explore later.

\[\sigma=\sqrt{\sigma^2}=\sqrt{\frac1N\sum_{i=1}^N(x_i-\mu)^2}\]

Generally, a standard deviation tells us how far away an average observation is from the mean.

Review: Sample Statistics and Parameters

To review, we have two main sample statistics (\(\bar{x},s^2\)) and two main population parameters (\(\mu,\sigma^2\)) to measure central tendency and spread, respectively. Generally, we give the mean and variance in parentheses for a given variable — remember to take the square root for standard deviation!

Spread: Coefficient of Variation

However, simply having higher standard deviation does not necessarily mean a variable has more variability relative to its mean or dispersion — its units could simply be larger.

For example, \(x=(5,64); y = (1,25)\). Does \(x\) or \(y\) have a higher level of dispersion?

\(\Rightarrow\) To answer this, we must introduce the coefficient of variation (CV), which weights our measure of spread by our measure of central tendency: \[CV=\frac{s}{\bar{x}}\]

Since \(\frac51>\frac85\), \(y\) has a higher level of dispersion, despite having a lower standard deviation.

Skewness

We also have more complicated summary statistics for our data:

Symmetry of our data means it can be perfectly reflected around the median (\(p^{50}\))
Data that has a tail to the right is right- or positive-skewed
Data that has a tail to the left is left- or negative-skewed

Three distribution curves labeled left-skewed, symmetric, and right-skewed. The left-skewed curve has a long tail extending to the left. The symmetric curve is bell-shaped with equal tails. The right-skewed curve has a long tail extending to the right.

Left-skewed, symmetric, and right-skewed distributions.

Measuring Skewness

We say our data is:

right-skewed if it has a skewness \(>0\)
left-skewed if its skewness \(<0\)
symmetric if its skewness \(\approx0\)

Skewness: Mean vs. Median

We can also use the comparison between our computed mean and median to impute skewness. What does it mean to have:

\(\bar{x}>median\) \(\Rightarrow\) the mean is pulled above the median by a long right tail: right-skewed
\(\bar{x}<median\) \(\Rightarrow\) the mean is pulled below the median by a long left tail: left-skewed
\(\bar{x}\approx median\) \(\Rightarrow\) no tail is pulling the mean away from the median: symmetric

Kurtosis

We will consider one more measure of our data, kurtosis. This is a measure of how fat the tails of our data are; how represented values far away from the mean are.

We define a normal distribution as one with symmetry/zero skewness (median=mean) and a kurtosis of exactly 3
Distributions with a kurtosis \(>3\) have fatter tails than the normal (“leptokurtic”), and those with kurtosis \(<3\) have thinner tails than the normal (“platykurtic”)¹.

Kurtosis: Graphical Representation

Density plot showing three bell-shaped curves with the same mean and variance, differing only in kurtosis. The purple solid curve is leptokurtic with kurtosis greater than 3, showing a taller sharper peak and heavier tails that decay slowly. The blue solid curve is the standard normal distribution with kurtosis equal to 3, the reference. The red dashed curve is platykurtic with kurtosis less than 3, showing a flatter peak and thinner tails that reach zero within a finite range. Labels on the right side of the plot identify each curve by kurtosis.

Density curves illustrating kurtosis: leptokurtic (\(\kappa > 3\)), normal (\(\kappa = 3\)), and platykurtic (\(\kappa < 3\)) distributions with equal mean, variance.

Putting It Together

Together, central tendency, spread, skewness, and kurtosis characterize datasets we will work with. Understanding how our dataset compares to a normal distribution will be important for the statistical inference we will conduct.

Data Visualization

Overview

We will now move to standard ways to visualize data we have collected. This often comes in the form of plots or summary tables. You will be asked to replicate many of these for example data in Stata for homework.

Summary Statistics Table

A summary statistics table gives statistics for a given sample of data:

(1978 automobile data)

                            Price
-------------------------------------------------------------
      Percentiles      Smallest
 1%         3291           3291
 5%         3748           3299
10%         3895           3667       Obs                  74
25%         4195           3748       Sum of wgt.          74

50%       5006.5                      Mean           6165.257
                        Largest       Std. dev.      2949.496
75%         6342          13466
90%        11385          13594       Variance        8699526
95%        13466          14500       Skewness       1.653434
99%        15906          15906       Kurtosis       4.819188

Box-and-Whisker Plot

A box[-and-whisker] plot gives the median, IQR, (skewness,) and any outliers for a given variable:

Box plot of automobile prices from the Stata auto dataset.

Frequency Table

A frequency table gives statistics for each value of a variable:

Frequency is how often a given value of a variable is represented in our data
Relative frequency (or percent) is this quantity scaled by the total frequency of all values
Cumulative percent is the proportion of total values made up by the given value and all previous/lower values

Frequency Table: Example

tabulate foreign


 Car origin |      Freq.     Percent        Cum.
------------+-----------------------------------
   Domestic |         52       70.27       70.27
    Foreign |         22       29.73      100.00
------------+-----------------------------------
      Total |         74      100.00

Histogram

A histogram plots frequency of data values organized into bins. As a researcher, it is often up to you to determine the optimal bin width for the data you are presenting. (Common default \(\sqrt{n}\))

Histogram of net worth data with a wide default bin width.

Kernel Density Plot

A kernel density plot is a smoothed version of the histogram that uses windows instead of bins. The weights given to values within a window are called kernel weights. The kernel density plot is a smooth curve that estimates the underlying distribution of the data, and can be overlaid with a normal distribution for comparison.

Kernel density estimate of net worth with a normal distribution.

Bar Charts and Pie Charts

Some additional charts:

Bar charts plot values for discrete categories instead of continuous bins or windows
Pie charts give each value as a relative frequency of a total

Data Transformations

Log Transformation

Sometimes we may need to apply a transformation to our data to have it appear more symmetric. One such transformation for right-skewed data is often the natural logarithm, which can reduce large outlying (right) values in right-skewed data.

Data that appears normal once log-transformed is said to follow a lognormal distribution.
We should not apply the same log transformation to left-skewed data, however, as this would only make the skew worse. Remember that we also cannot take a log of a negative number.

Log Transformation: Example

Histogram of log net worth values with a normal distribution curve overlaid. The log-transformed distribution appears approximately symmetric and close to normal, contrasting with the right-skewed untransformed net worth data.

Histogram of log-transformed net worth with a normal distribution.

Standardization and Z-Scores

We may also want to standardize our data and transform it into a z-score. To do this, we subtract the population mean and divide by the population standard deviation: \[z_i=\frac{x_i-\mu}{\sigma}\]

Note: z-scores will always have \((0,1)\) as a result but will not magically become normally distributed. A non-normal standardized variable will remain non-normal even as \(n\rightarrow\infty\). A distribution that is normal will remain normal after standardizing.

Interpreting Z-Scores

Standardized [z-]scores allow us to compare values from series that are scaled differently, such as two different exams with different means and standard deviations. A 1-unit change (\(\pm1\)) in a standardized variable \(z_x\) will always correspond to \(\pm\sigma\) for variable \(x\).

If you scored 80 on an exam \(\sim(65,100)\) and 62 on an exam \(\sim(50,36)\), on which exam did you perform better relative to the average (with notation \(X\sim(\mu,\sigma^2)\) for mean \(\mu\) and variance \(\sigma^2\))?

We must use z-scores to standardize the two exams and compare your performance relative to the average for each exam. \[z_1=\frac{80-65}{\sqrt{100}}=1.5\hspace{1cm}z_2=\frac{62-50}{\sqrt{36}}=2\]

Despite scoring higher on the first exam, you performed better relative to the average on the second exam, since \(z_2>z_1\).

Growth Rates

It may be more interesting to analyze the growth rates of a variable than the variable itself (or even, growth rates of growth rates, e.g. inflation). To compute a growth rate for a variable \(x_t\): \[\%\Delta x_t=100\times\frac{x_t-x_{t-1}}{x_{t-1}}\text{ or }100\times\left(\frac{x_t}{x_{t-1}}-1\right)\]

We may need to annualize this data if it does not come from an annual-frequency series already. For quarterly data (e.g. GDP), this would mean multiplying the growth rate by 4.

End of Lecture Material

Knowledge Check 2

Suppose we compute \(\bar{x}=2.5, Median=1.5\) for a sample of data:

Without using formulas, is this data symmetric or skewed? Is it normally distributed? Why or why not?
Would you expect the z-scores for these observations to be normally distributed? Why or why not?