3. Numerical Summaries and Comparing Distributions Free Online FlashCards

What four features describe a quantitative distribution?

Center, spread, position, and shape. A single numerical summary cannot reveal every important feature, so summaries should be interpreted with an appropriate graph.

How is the mean calculated?

The mean is the arithmetic average: x̄ = Σxᵢ/n. For 4, 5, 6, 7, and 28, the mean is 10.

What is the median?

The median is the middle ordered observation. With an even number of observations, it is the average of the two middle values.

Which center-and-spread pair suits skewed data?

Use the median with the IQR when data are skewed or contain outliers. Use the mean with standard deviation when data are reasonably symmetric without influential outliers.

How is the range calculated, and what limits it?

The range is maximum − minimum. It is easy to calculate but can be misleadingly large because it depends only on the two extreme observations.

What does the IQR measure?

IQR = Q₃ − Q₁. It measures the spread of the middle 50% of observations and is resistant to outliers.

What does sample standard deviation measure?

A sample standard deviation measures typical distance from the sample mean: s = √[Σ(xᵢ − x̄)²/(n − 1)]. Its units match the original variable.

What is the empirical rule?

For an approximately bell-shaped distribution, about 68% lie within 1 standard deviation, 95% within 2, and 99.7% within 3. The rule does not automatically apply to strongly skewed data.

What does the p-th percentile indicate?

The p-th percentile is a value at or below which approximately p% of observations fall. The 50th percentile is the median.

What values make up the five-number summary?

Minimum, Q₁, median, Q₃, and maximum. These five values provide a compact description of location and spread and form the basis of a boxplot.

What does a z-score represent?

A z-score gives the number of standard deviations an observation is from the mean: z = (x − x̄)/s for a sample. A positive value is above the mean; a negative value is below it.

How does the 1.5-IQR rule flag possible outliers?

Lower fence = Q₁ − 1.5(IQR); upper fence = Q₃ + 1.5(IQR). Values beyond these fences are flagged as potential outliers, not proven errors.