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 or data with outliers.

Mode:

• • The most frequently occurring value or values.
  • Useful mainly for categorical or discrete data; not as common in AP Stats analysis of quantitative data.

Measures of Spread (Variability)

Range:

\( \text{Range} = \text{Maximum} – \text{Minimum} \)

• • Simple, but influenced by outliers.

Interquartile Range (IQR):

\( IQR = Q_3 – Q_1 \)

• • Measures the spread of the middle 50% of the data.
  • Resistant to outliers.

Variance:

\( s^2 = \dfrac{\sum (x_i – \bar{x})^2}{n-1} \)

• • Average squared deviation from the mean.

Standard Deviation:

\( s = \sqrt{ \dfrac{\sum (x_i – \bar{x})^2}{n-1} } \)

• • Measures typical distance of data points from the mean.
  • Sensitive to outliers and skewness.

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.

Important Note

When describing data:

• • Use mean and standard deviation for symmetric distributions without outliers.
  • Use median and IQR for skewed distributions or when outliers are present.