2.6 Skewness and the Mean, Median, and Mode

Distribution Shapes and Skewness

Understanding skewness tells you how data is distributed around the center. This matters because the shape of a distribution determines which measure of central tendency (mean, median, or mode) best represents your data, and choosing the wrong one can lead to misleading conclusions.

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Symmetrical vs. Skewed Distributions

A symmetrical distribution has left and right sides that mirror each other. The classic example is the bell-shaped normal distribution, where data clusters evenly around the center.

A skewed distribution has a longer tail on one side:

• Right-skewed (positively skewed): The tail stretches to the right. Most data points cluster on the left side. A common example is household income, where most people earn moderate amounts but a few very high earners pull the tail rightward.
• Left-skewed (negatively skewed): The tail stretches to the left. Most data points cluster on the right side. Think of exam scores on an easy test, where most students score high but a few very low scores pull the tail leftward.

Kurtosis is a separate concept that measures how heavy or light the tails are compared to a normal distribution. High kurtosis means more data in the tails (and a sharper peak), while low kurtosis means thinner tails (and a flatter peak). Skewness and kurtosis describe different aspects of shape.

Positions of Mean, Median, and Mode

The relationship between mean, median, and mode shifts depending on the distribution's shape.

Symmetrical distribution: $\text{Mean} = \text{Median} = \text{Mode}$ All three sit together at the center.

Right-skewed distribution:

1. Mode sits at the peak (furthest left)
2. Median falls between the mode and the mean
3. Mean is pulled furthest to the right by extreme high values

So the order is: Mode < Median < Mean.

Left-skewed distribution:

1. Mean is pulled furthest to the left by extreme low values
2. Median falls between the mean and the mode
3. Mode sits at the peak (furthest right)

So the order is: Mean < Median < Mode.

The mean always gets "dragged" toward the tail because it's sensitive to every value in the dataset, including outliers. The median and mode are more resistant to that pull.

Impact of Skewness on Measures of Central Tendency

When data is skewed, the mean and median tell different stories. Knowing which to trust is one of the most practical takeaways from this topic.

• Right-skewed: Mean > Median. The mean is inflated by high outliers, so the median better represents the typical value. For example, if you report average household income, a few billionaires drag that number up. Median income gives a more honest picture of what a typical household earns.
• Left-skewed: Mean < Median. The mean is deflated by low outliers, so again the median is the more robust choice.
• Symmetrical: Mean ≈ Median ≈ Mode. No skew is pulling the mean in either direction, so all three measures work well for describing the center.

Descriptive Statistics and Distribution Characteristics

These concepts connect to the bigger picture of descriptive statistics:

• Measures of central tendency (mean, median, mode) describe where the data tends to cluster, giving you a single "typical" value.
• Measures of dispersion (range, standard deviation, IQR) describe how spread out the data is around that center. Two datasets can have the same mean but very different spreads.
• The probability density function (PDF) describes the relative likelihood of different values in a continuous distribution. The area under the curve between two values gives the probability of a data point falling in that range, and the total area under the curve always equals 1.

Skewness ties these ideas together: it tells you how the data spreads asymmetrically, which in turn tells you which central tendency measure to report and how to interpret the variability you observe.