Summary Statistics for a Quantitative Variable - AP Statistics

Range = 20. Maximum $30$ minus minimum $10$ equals $20$.

Median = 6. Ordered: $3, 5, 6, 7, 8$; middle value is $6$.

Range = 20. Maximum $30$ minus minimum $10$ equals $20$.

Median = 6. Ordered: $3, 5, 6, 7, 8$; middle value is $6$.

$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$. Each value is multiplied by its weight before averaging.

Minimum, Q1, Median, Q3, Maximum. Five key values that describe a dataset's distribution.

A histogram displays the frequency distribution of a dataset. Shows data distribution using bars for frequency counts.

$s_{xy} = \frac{1}{n-1} \sum (x_i - \bar{x})(y_i - \bar{y})$. Measures how two variables change together linearly.

Range = 7. Maximum $9$ minus minimum $2$ equals $7$.

IQR = 5. $Q^3 = 8$, $Q^1 = 3$, so $IQR = 8 - 3 = 5$.

Heavier tails than a normal distribution. More extreme values in tails than normal distribution.

An outlier is a data point significantly different from others. Falls unusually far from other observations in the dataset.

A percentile indicates the relative standing of a value in a dataset. Shows what percentage of data falls below a given value.

$s = 0$. All values are identical, so no variability exists.

$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$. Average squared deviation from sample mean using $n-1$.

Skewness = 0. Symmetric distributions have equal left and right tails.

$\bar{x} = 6$. Sum is $30$, divided by $5$ observations gives $6$.

$z = \frac{x - \bar{x}}{s}$. Number of standard deviations a value is from the mean.

$CV = \frac{s}{\bar{x}} \times 100%$. Relative variability as a percentage of the mean.

Mode = 4. Value $4$ appears three times, more than any other.

60th percentile. Three out of five values are below $4$.

Kurtosis measures the 'tailedness' of the distribution. High kurtosis means heavy tails; low means light tails.

Median = 3. Ordered: $1, 1, 3, 4, 5$; middle value is $3$.

$s^2 = 2.5$. Mean is $3$; variance is $\frac{10}{4} = 2.5$.

IQR = Q3 - Q1. Distance between third quartile and first quartile.

Range = Maximum value - Minimum value. Difference between highest and lowest values in the dataset.

A boxplot displays the distribution through five-number summary. Shows quartiles, median, and potential outliers graphically.

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$. Sum all values and divide by sample size $n$.

$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$. Square root of sample variance using $n-1$ in denominator.