Data Visualization – Ready reference – Do the math

The mean, or average, uses every value in a data set. It works well when each observation should count the same. But very high or low values can pull it away from what is typical.

The median shows the middle of an ordered data set. Because extreme values have less effect on it, it is useful for uneven data, such as home prices or household incomes.

The mode shows the most common value or category. Use it when you want to show which result appears most often.

Definitions

The mean is found by adding all values and dividing by the number of data points.

The median is the middle value when the data is in numerical order. If the data set has an even number of observations, find the median by averaging the two middle values.

The mode is the value that appears most often. A data set can have more than one mode if several values share the top count.

Example: The following data set shows the number of cars sold in a day by each of nine divisions.

A symmetric distribution has the same shape on both sides of its center. Some symmetric distributions follow a bell shape, as test scores sometimes do.

Example: Roll two fair, six-sided dice many times and plot the sum of each roll. Over many trials, the results move toward a symmetric, triangular distribution centered on 7. Sums near the center occur more often than those at either end.

Fat tails, often tied to high kurtosis, mean that extreme gains or losses happen more often than a normal bell curve would suggest. Although it is widely used, the normal distribution can downplay the chance of extreme events, such as the 1987 stock market crash.

Skewed distributions show data that is spread unevenly, with a longer tail on one side. Common examples include personal income and losses from credit defaults.

Personal income is a well-studied example of a skewed distribution: most people earn in the lower or middle range, while a smaller group earns much more. Researchers study this pattern to understand the income gap, poverty, and economic growth.

Even bonds with a low chance of default carry some risk of losing part or all of the invested principal during the holding period.

A risk-taker focuses on the remote chance of winning the lottery. A risk-averse person worries about an equally unlikely danger, like someone striking them with lightning.

Both give weight to rare, extreme outcomes—the events found in the tails of a probability distribution.

A logarithmic scale works across a wide range of values. Equal spacing shows equal ratios, not equal gaps.

When the focus is on the near term, a log scale on the time axis makes shorter spans easier to tell apart while keeping more distant periods visible.

Example: A yield curve showing Treasury bills, notes, and bonds across different maturities.

A logarithmic y-axis shows proportional changes instead of absolute differences. Equal percent increases take up the same vertical space, no matter the starting value.

Example: A 10-point rise from 10 to 20 is a 100% gain, while a rise from 100 to 110 is 10%. From 1,000 to 1,010, the gain is just 1%. A logarithmic scale shows these differences with smaller and smaller vertical moves.

A logarithmic y-axis is especially useful for showing stock prices or market indexes over long periods.

Logarithmic scales can use different bases, but base 10 is a practical and easy-to-read choice for most charts.

log_{10}10 = 1 log_{10}50 = 1.7 log_{10}100 = 2 log_{10}500 = 2.7 log_{10}1000 = 3 log_{10}5000 = 3.7 log_{10}10000 = 4

Stock P trades between $21 and $24, while Stock Q trades between $105 and $110.

Both graphs now use comparable scales, making the two stocks’ relative performance easy to compare.

Use the mean when every value in the data set should count equally and the data is not heavily skewed by very high or low values. Use the median when the data is unevenly spread or has outliers, such as household incomes or home prices, because the median is less affected by extreme values.

Standard deviation helps show how much returns vary around their average. In finance, that spread is often read as volatility: a higher standard deviation means returns are more spread out, and the asset's price may be less predictable. For daily returns, annualized volatility is often estimated by multiplying the daily standard deviation by the square root of the number of trading days in a year.

A simple average gives every observation the same weight, while a weighted average gives more weight to some observations based on things such as trading volume, duration, or money value. For example, a volume-weighted average price gives more importance to prices at which more shares were traded, making it a better match for actual market activity than a simple average of recorded prices.

Logarithmic scales are useful when values span a wide range or when proportional change matters more than absolute change. On a logarithmic y-axis, equal percent increases appear as equal vertical distances, which makes growth rates easier to compare over time. On a logarithmic x-axis, shorter time spans can be shown in more detail while longer spans stay visible.

Re-indexing sets the first value of a series to 100, which makes later changes easier to read as percentage gains or losses from the starting point. For example, a move from 100 to 113 clearly shows a 13% increase. This is especially useful when you compare series that begin at different absolute values.