THE GARDEN

Descriptive statistics and graphs

Updated 11 Oct 2026

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  Frequency tables

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    Frequency distribution -> groups values or classes and counts observations.

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    Class limits -> stated endpoints.

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    Class boundaries -> separate adjacent intervals without gaps.

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    Class width -> interval span.

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    Class midpoint -> representative value for approximate calculations.

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    Class checks

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      Classes do not overlap.

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      Every observation fits a class.

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      Frequencies sum to $n$.

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    Relative frequency -> $\frac{f_i}{n}$.

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    Cumulative frequency -> adds counts through each ordered class.

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  Choosing classes

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    Sturges’ rule, $k\approx1+3.322\log_{10}(n)$, is a starting heuristic, not a requirement.

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    grouping choices can change the apparent shape of a distribution; report the choice.

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  Graphs

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    Bar chart -> categories; bars have gaps.

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    Histogram -> numeric intervals; adjacent bars.

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    Line chart -> ordered sequence, often time.

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    Pie chart -> parts of one whole; easiest to read with few categories.

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    Frequency polygon -> connects class-midpoint frequencies.

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    Ogive -> displays cumulative frequency.

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    Stem-and-leaf plot -> preserves individual values in a modest dataset.

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    Box plot -> summarizes median, quartiles, and potential outliers.

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    Broken axis caution -> can exaggerate differences if the break is easy to miss.

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    Add a compact comparison image: bar chart versus histogram, plus a box plot and ogive with the same small dataset.

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  Worked grouped example

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    Vehicle-price slide frequencies -> $8,23,17,18,8,4,2$; total $n=80$.

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    Relative frequencies -> $0.10,0.2875,0.2125,0.225,0.10,0.05,0.025$.

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    Cumulative counts -> $8,31,48,66,74,78,80$.

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  Center

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    Mean -> $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$; sensitive to extreme values.

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    Median -> middle ordered value, or the mean of the two middle values; more resistant to skew and outliers.

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    Mode -> most frequent value or category; a dataset may have several modes or no useful mode.

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  Spread

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    Range -> maximum minus minimum.

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    Population variance -> $\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2$.

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    Sample variance -> $s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2$ under the usual independent-sampling setup.

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    Standard deviation -> square root of variance and uses the same units as the data.

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    Mean deviation summarizes absolute distances from a chosen center.

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    the $n-1$ divisor is Bessel’s correction for the usual unbiased sample-variance estimator, not a universal divisor for every dataset.

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  Position and relative spread

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    quartiles and percentiles describe positions in ordered data; exact values may vary with the interpolation convention.

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    Quartile deviation -> commonly half the interquartile range, $\frac{Q_3-Q_1}{2}$.

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    Coefficient of variation -> $\frac{s}{\bar{x}}\times100\%$; use only for ratio-scale data with a nonzero mean.

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  Grouped-data estimates

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    with class midpoint $m_i$ and frequency $f_i$, mean is approximately $\frac{\sum_i f_i m_i}{n}$.

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    this is approximate because it substitutes class midpoints for original observations.

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    grouped median and mode formulas interpolate within a class and depend on their assumptions.

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