Descriptive Statistics and Numerical Methods

The Median (The Middle)

The median ($Md$) is the middlemost value in a data set when the observations are arranged in increasing order.

• If the number of observations is odd, the median is the exact middle value at position $(n+1)/2$.
• If the number of observations is even, the median is the mean of the two middle values.

The Mode (The Peak)

The mode ($Mo$) is the value that occurs most frequently in a data set. Unlike the mean and median, a data set can have more than one mode:

• Unimodal: One clear peak.
• Bimodal: Two distinct peaks.
• Multimodal: Three or more peaks.

Outliers and Resistant Measures

Not all measures of center are created equal. The mean is very sensitive to outliers—extreme data values that are far from the rest of the observations. For example, if you are looking at NBA salaries, a few players earning over $33 million can pull the "average" (mean) much higher than what a typical player actually earns.

Because of this sensitivity, we distinguish between "sensitive" and "resistant measures":

• The Mean is NOT resistant: A single massive outlier can drastically change the mean.
• The Median and Mode ARE resistant: The median is only affected by the position of an extreme value, not its specific magnitude. The mode is only sensitive to the most frequent value and ignores everything else.

When a distribution is heavily skewed (having a very long tail to the left or right), the median is often a better representation of the "typical" value than the mean.

Measuring Variation and Spread

Knowing the center is only half the battle; we also need to know how spread out the data is. This is known as variation.

Variance and Standard Deviation

The population variance ($\sigma^2$) is the average of the squared deviations from the mean: $\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$ The population standard deviation ($\sigma$) is simply the positive square root of the variance: $\sigma = \sqrt{\sigma^2}$

The Empirical Rule and Z-Scores

If a population follows a normal (bell-shaped) curve, we can use the Empirical Rule to predict where data falls:

• Approximately 68.2% of measurements lie within $\mu \pm 1\sigma$.
• Approximately 95.4% of measurements lie within $\mu \pm 2\sigma$.
• Approximately 99.7% of measurements lie within $\mu \pm 3\sigma$.

To find out exactly how many standard deviations a specific value is from the mean, we calculate a z-score: $z = \frac{x - \mu}{\sigma}$ A z-score above $+3$ or below $-3$ is often categorized as an outlier.

Visualizing Data: Percentiles and Box Plots

To summarize data visually, we often use the five-number summary, which includes the Minimum, First Quartile ($Q_1$), Median, Third Quartile ($Q_3$), and Maximum.

• Percentiles: The $p^{th}$ percentile is the value below which at least $p$ percent of the observations fall.
• Quartiles: $Q_1$ is the 25th percentile, and $Q_3$ is the 75th percentile.
• Interquartile Range (IQR): This measures the spread of the middle 50% of the data: $IQR = Q_3 - Q_1$.

The Box-and-Whiskers Display

A box plot (or box-and-whisker display) is a graphical tool that represents this five-number summary:

1. The box represents the area between $Q_1$ and $Q_3$.
2. A line inside the box marks the median.
3. The whiskers extend to the minimum and maximum values.

Warning: When drawing box plots, if a data point is more than $1.5 \times IQR$ away from the quartiles, it should be plotted as a separate dot to indicate it is a suspected outlier.

Summary Checklist

When analyzing a new data set, follow these best practices:

1. Always plot your data first (using a histogram or dot plot) to see the shape and identify outliers.
2. Check for symmetry: Is it mound-shaped (normal) or skewed?
3. Identify peaks: Is it unimodal, bimodal, or multimodal?
4. Choose your measures: Use the mean for symmetric data, but consider the median if outliers are present.