AP Statistics 1.7 Summary Statistics for a Quantitative Variable Study Notes

Summary Statistics for a Quantitative Variable

To describe a quantitative distribution numerically, we use measures of center, spread (variability), and position. These are called summary statistics.

Measures of Center

Mean (Arithmetic Average):

\( \displaystyle \bar{x} = \dfrac{\text{sum of data values}}{\text{number of values}} \)

• Sensitive to outliers and skewed data.
• Best for symmetric distributions with no extreme values.

Median:

• The middle value when data are ordered.
• If even number of observations → average of the two middle values.
• Resistant to outliers and skewness.
• Best for skewed data or when outliers are present.

Mode:

• The most frequently occurring value(s).
• Useful mainly for categorical or discrete data; less common in quantitative analysis.

Measures of Position

Quartiles:

• \( Q_1 \) = median of lower half (25th percentile).
• \( Q_3 \) = median of upper half (75th percentile).

Percentiles:

The \(k\)-th percentile is a value where \(k\%\) of the data are at or below it.

Z-Score (Standardized Score):

\( z = \dfrac{x – \bar{x}}{s} \)

• Tells how many standard deviations a value is from the mean.
• Positive \(z\) → above mean; Negative \(z\) → below mean.
• Helps compare values from different distributions.