Center and Spread

Learn arithmetic mean, population standard deviation.

Lesson 1 of 32 parts Practice Flashcards

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Arithmetic Mean

Objective: Explain arithmetic mean, apply the stated method, and check its conditions.

The arithmetic mean is the sum of numerical observations divided by their count. It uses every value and can be shifted by an outlier. A mean need not be an observed value, and it should be reported in the same units as the data. An extreme value can pull the mean away from most observations.

Formulas

$x̄=(Σxᵢ)/n$

Worked examples

Try itFind the mean of 1, 7, and 13.Show answer

Step 1: Add the three observations: 21. Step 2: Divide by the number of observations, 3. Step 3: The mean is 7.

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Arithmetic Mean

Objective: Explain arithmetic mean, apply the stated method, and check its conditions.

Formulas

$x̄=(Σxᵢ)/n$

Worked examples

Try itFind the mean of 1, 7, and 13.Show answer

Step 1: Add the three observations: 21. Step 2: Divide by the number of observations, 3. Step 3: The mean is 7.

Population Standard Deviation

Objective: Explain population standard deviation, apply the stated method, and check its conditions.

Population standard deviation is the square root of the mean squared deviation from the population mean. It uses denominator N for a complete population; a sample variance estimate typically uses n−1. SD is nonnegative and has the data’s original units. Use population N only when treating the data as the complete population.

Formulas

$σ=√[Σ(xᵢ−μ)²/N]$

Worked examples

Try itTreat the two values 1 and 13 as the entire population. Find the population standard deviation.Show answer

Step 1: The population mean is 7. Step 2: Squared deviations are 36 and 36; their population mean is 36. Step 3: Take the square root to get 6.

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