2.6 Skewness and the Mean, Median, and Mode

Characteristics and Relationships of Distributions

Symmetrical vs. skewed distributions

The shape of a distribution tells you a lot about your data before you even calculate anything. Recognizing whether a distribution is symmetrical or skewed helps you decide which measure of central tendency best represents your data.

• Symmetrical distributions have a bell-shaped curve where the mean, median, and mode are all equal and located at the center. The two halves of the distribution are mirror images of each other. The classic normal distribution is the most common example.
• Right-skewed (positively skewed) distributions have a tail that extends further to the right. A few unusually large values pull the tail out in that direction. Income distribution is a classic example: most people earn moderate amounts, but a small number of very high earners stretch the right tail.
  • The ordering is $\text{Mean} > \text{Median} > \text{Mode}$
• Left-skewed (negatively skewed) distributions have a tail that extends further to the left. Here, a few unusually small values pull the tail to the left. Think of age at retirement in a company: most people retire around 65, but a few retire very young, pulling the left tail out.
  • The ordering is $\text{Mode} > \text{Median} > \text{Mean}$
• Kurtosis measures how peaked or flat a distribution is compared to a normal distribution. High kurtosis means a sharper peak and heavier tails; low kurtosis means a flatter peak and lighter tails. For an intro course, just know it describes the "shape" beyond skewness.

Mean, median, and mode relationships

The key idea here is that the mean gets pulled toward the tail of a skewed distribution. The median resists that pull, and the mode stays at the peak.

• In a symmetrical distribution: $\text{Mean} = \text{Median} = \text{Mode}$. All three sit together at the center.
• In a right-skewed distribution: $\text{Mean} > \text{Median} > \text{Mode}$. The mean chases the long right tail.
• In a left-skewed distribution: $\text{Mode} > \text{Median} > \text{Mean}$. The mean chases the long left tail.

A quick way to remember: the mean always gets dragged toward the tail, the mode always stays at the highest point of the curve, and the median falls in between.

Impact of Outliers on Measures of Central Tendency

Outlier effects on central tendency

Outliers are extreme values that sit far from the rest of your data. They don't affect all measures of central tendency equally.

• The mean is the most sensitive to outliers. Because the mean uses every value in its calculation, even one extreme number can shift it substantially.
  • In a right-skewed distribution, high outliers (like a CEO's salary in a dataset of employee incomes) pull the mean to the right.
  • In a left-skewed distribution, low outliers (like a student scoring a 12 on an exam where most scored 80+) pull the mean to the left.
• The median is resistant to outliers. It depends on the position of values, not their size. If you have 100 data points, the median is the average of the 50th and 51st values regardless of how extreme the largest or smallest values are. That's why median household income is often reported instead of mean household income.
• The mode is unaffected by outliers. It simply identifies the most frequently occurring value. Since outliers are rare by definition, they don't change which value appears most often.

Measures of Variability

Variability tells you how spread out your data is. Two datasets can have the same mean but look completely different if one is tightly clustered and the other is widely spread.

• Standard deviation measures the average distance between each data point and the mean. A small standard deviation means data points are clustered close to the mean; a large one means they're spread out.
• Variance is the standard deviation squared ($\text{Variance} = \sigma^2$). It represents the same idea as standard deviation but in squared units, which makes it less intuitive to interpret directly. It's used more often in calculations behind the scenes.
• Interquartile range (IQR) is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$): $\text{IQR} = Q_3 - Q_1$. It captures the middle 50% of your data. Because it ignores the extremes, the IQR is much less affected by outliers than the standard deviation.

Just as the median resists outliers better than the mean, the IQR resists outliers better than the standard deviation. When your data is skewed, report the median and IQR together. When your data is symmetrical, the mean and standard deviation are the better pair.

Histograms are the go-to visual tool for spotting skewness. By looking at the shape of a histogram, you can quickly tell whether a distribution is symmetrical, right-skewed, or left-skewed before doing any calculations.