Mean, median, and mode

Key idea

The mean (average) uses every value: \[ \bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}. \] The median is the middle value after sorting the data (or the average of the two middle values when \(n\) is even). The mode is the most frequent value (data can be bimodal or have no mode). In skewed data or with outliers, the median is usually more resistant than the mean.

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Summary

• Mean uses every value, but can be pulled by outliers.
• Median is the middle of ordered data and is resistant to outliers.
• Mode describes the most frequent value(s), especially useful for categories.

Range, quartiles, and interquartile range (IQR)

Key idea

The range is the simplest measure of spread: \[ \text{Range}=\text{max}-\text{min}. \] Quartiles divide ordered data into four parts. The interquartile range is: \[ \text{IQR}=Q_3-Q_1, \] which measures the spread of the middle 50% of the data (and is less affected by outliers than the range).

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Summary

• Range measures overall spread, but is sensitive to extreme values.
• IQR measures spread of the middle 50% and is more resistant to outliers.

Frequency, relative frequency, and percent

Key idea

Frequency is a count. Relative frequency is the fraction of the total: \[ \text{Relative frequency}=\frac{\text{count}}{n}. \] To convert to a percent, multiply by 100: \[ \text{Percent}= \left(\frac{\text{count}}{n}\right)\cdot 100\%. \] This helps you compare groups even when sample sizes are different.

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Summary

• Relative frequency is “part of the whole”: \(\frac{\text{count}}{n}\).
• Percent is relative frequency \(\times 100\%\).

Five-number summary and box-and-whisker plots

Key idea

A five-number summary describes a data set using: \[ \text{min},\ Q_1,\ \text{median }(Q_2),\ Q_3,\ \text{max}. \] A box-and-whisker plot uses these five numbers: the box runs from \(Q_1\) to \(Q_3\), the median is a line inside the box, and the whiskers extend toward the minimum and maximum (or toward the non-outlier values, depending on the convention).

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Summary

• Five-number summary: min, \(Q_1\), median, \(Q_3\), max.
• A box plot visualizes the middle 50% (the box) and the overall spread (the whiskers).

Outliers, the 1.5×IQR rule, and midrange

Key idea

A common outlier check uses IQR fences: \[ \text{Lower fence}=Q_1-1.5(\text{IQR}), \quad \text{Upper fence}=Q_3+1.5(\text{IQR}). \] Values outside the fences are often treated as outliers. Another measure you may see is the midrange, the midpoint of the minimum and maximum: \[ \text{Midrange}=\frac{\text{min}+\text{max}}{2}. \] Midrange is easy to compute, but it depends only on the extremes (so it is not resistant to outliers).

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Summary

• Outlier fences: \(Q_1-1.5(\text{IQR})\) and \(Q_3+1.5(\text{IQR})\).
• Midrange is easy but depends only on the extremes.

Variance, standard deviation, and z-scores

Key idea

Variance measures average squared distance from the mean, and standard deviation is its square root. For a population (using all values), one definition is: \[ \sigma^2=\frac{\sum (x-\mu)^2}{n}, \quad \sigma=\sqrt{\sigma^2}. \] A z-score measures how many standard deviations a value is from the mean: \[ z=\frac{x-\mu}{\sigma}. \]

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Summary

• Standard deviation describes a typical distance from the mean.
• Z-scores compare values across different scales by measuring “how many standard deviations away.”

Choosing the right statistic and interpreting data

Where descriptive statistics show up

• School and testing: summarize scores with mean/median and compare spread with IQR.
• Science and experiments: report “typical result” and variability using standard deviation.
• Sports and performance: compare consistency (low spread) vs volatility (high spread).
• Data literacy: interpret charts like box plots and frequency tables in reports.

Worked example: median from unsorted data

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Final recap

• Center: mean, median, mode describe typical value in different ways.
• Spread: range and IQR describe variability; IQR is more resistant to outliers.
• Five-number summary and box plots summarize data visually and support outlier checks.
• Standard deviation measures typical distance from the mean; z-scores compare across scales.