Key takeaway: The standard deviation measures how far data values typically lie from the mean, so a small value means tightly clustered data and a large one means widely spread data.
Grade 9 – Algebra 1S-ID.A.2Statistics and Data
Definition of Standard Deviation
The standard deviation describes spread in a single number, in the same units as the data. Two classes can have the same mean test score but very different consistency: one class with all scores near 80, another with scores from 40 to 100. The second has the larger standard deviation.
To find it, measure how far each value is from the mean, square those distances so that negatives do not cancel positives, average the squares, and take the square root. The average of the squares is the variance. A value near zero means that nearly everything equals the mean. Statisticians use the population version when the data are the whole group and the sample version, dividing by \(n – 1\), when the data are a sample.
Formula and notation
For a whole population of \(N\) values with mean \(\mu\):
\[\sigma = \sqrt{\frac{\sum (x – \mu)^{2}}{N}}\]
For a sample of \(n\) values with mean \(\,\overline{x}\), the symbol is \(s\) and the divisor is \(n – 1\). Calculators and spreadsheets have buttons for both.
Key facts
- Standard deviation is never negative, and it is zero only if every value is identical.
- It has the same units as the data, unlike the variance.
- Adding the same number to every value does not change the spread.
- A few extreme values can raise it sharply.
Standard Deviation example
Cassidy’s bus arrived 6, 8, 9, 12 and 15 minutes after she reached the stop. Find the population standard deviation of these waiting times .
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Mean: \(\frac{6 + 8 + 9 + 12 + 15}{5} = \frac{50}{5} = 10\).
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Deviations: \(-4, -2, -1, 2, 5\). Squares: \(16, 4, 1, 4, 25\), which total 50.
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Variance: \(\frac{50}{5} = 10\). Standard deviation: \(\sqrt{10} \approx 3.16\).
The standard deviation is about 3.16 minutes.
Common mistake
Watch out
Omar adds the deviations without squaring: \(-4 – 2 – 1 + 2 + 5 = 0\), and concludes that the data have no spread.
Why it is wrong
The deviations from the mean always cancel to zero, so they cannot measure spread directly. Squaring makes every term positive: \(16 + 4 + 1 + 4 + 25 = 50\). The data clearly vary, with a standard deviation of \(\sqrt{10} \approx 3.16\).
Related terms and lessons
- Mean Absolute Deviation (MAD)
- Range (Statistics)
- Interquartile Range (IQR)
- Outlier
- Correlation Coefficient
- Box Plot
- Algebra 1 · Statistics: Center, Spread and Data DisplaysLessonWorksheetQuiz
- AP Statistics · Exploring One-Variable DataLessonWorksheetQuiz
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