Mathematics
Grade 11
15 min
Variance and standard deviation
Variance and standard deviation
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define and differentiate between measures of central tendency (mean) and measures of dispersion (variance, standard deviation).
Calculate the variance and standard deviation for a given population dataset.
Calculate the variance and standard deviation for a given sample dataset, correctly applying Bessel's correction (n-1).
Interpret the meaning of standard deviation in the context of data consistency and spread.
Compare two datasets by analyzing their respective standard deviations.
Identify and avoid common errors in calculation, such as mixing up population and sample formulas.
Two archers shoot at a target. Both have the same average score, but one archer's arrows are tightly clustered while the other's are spread all over. Which archer...
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Key Concepts & Vocabulary
TermDefinitionExample
Mean (μ for population, x̄ for sample)The average of all the data points in a set. It's a measure of central tendency.For the dataset {2, 4, 4, 4, 5, 5, 7, 9}, the mean is (2+4+4+4+5+5+7+9) / 8 = 5.
DeviationThe difference between a single data point and the mean of the dataset. It shows how far an individual value is from the average.In the dataset {2, 4, 9} with a mean of 5, the deviation for the data point 9 is (9 - 5) = 4.
Variance (σ² for population, s² for sample)The average of the squared deviations from the mean. It measures the overall spread of the data, but its units are squared (e.g., cm²), which can be hard to interpret directly.If the squared deviations are {9, 1, 4}, the variance is (9+1+4)/3 = 4.67.
Standard Deviation (σ for population, s for sam...
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Core Formulas
Population Variance (σ²)
\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}
Use this formula when your dataset includes every member of the entire group you are studying. Here, μ is the population mean, xᵢ is each data point, and N is the total number of data points in the population.
Population Standard Deviation (σ)
\sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}
This is simply the square root of the population variance. It measures the typical distance of a data point from the population mean.
Sample Variance (s²)
s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}
Use this when your dataset is a sample (a subset) of a larger population. Here, x̄ is the sample mean, xᵢ is each data point, and n is the number of data points in the sample. Dividing by (n-1) is cal...
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Challenging
The use of (n-1) in the denominator of the sample variance formula is known as Bessel's correction. What is the primary statistical reason for this correction?
A.To make the calculation easier by using a smaller number.
B.To produce an unbiased estimate of the population variance.
C.To account for the possibility of negative data points.
D.To ensure the sample standard deviation is always an integer.
Challenging
For a population of 50 students, the sum of their test scores is Σxᵢ = 4000, and the sum of the squares of their scores is Σxᵢ² = 325,000. What is the population standard deviation?
A.10
B.100
C.80
D.50
Challenging
A sample of data has n=10 items and a sample standard deviation of s=4. What is the sum of the squared deviations from the mean, Σ(xᵢ - x̄)²?
A.160
B.16
C.144
D.40
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