Population & sample SD • 2026 edition
Population SD: \( \sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}} \)
Sample SD: \( s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \)
Variance: \( \sigma^2 = \frac{\sum(x_i - \mu)^2}{N} \) or \( s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1} \)
Mean: \( \mu = \frac{\sum x_i}{N} \) or \( \bar{x} = \frac{\sum x_i}{n} \)
Z-Score: \( z = \frac{x - \mu}{\sigma} \)
Standard deviation measures the spread or dispersion of data points from the mean. A low standard deviation indicates values are close to the mean, while a high standard deviation indicates values are spread out. The population formula divides by N, while the sample formula divides by n-1 (Bessel's correction).
Standard deviation measures the amount of variation or dispersion in a dataset. It quantifies how spread out values are from the mean. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.
\( \sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}} \)
Where σ is the population standard deviation, xi represents each data value, μ is the population mean, and N is the total number of values in the population.
\( s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}} \)
Where s is the sample standard deviation, xi represents each data value, x̄ is the sample mean, and n is the number of values in the sample. The n-1 in the denominator is Bessel's correction.
What does a standard deviation of 0 indicate about a dataset?
The answer is B) All values are the same. Standard deviation measures the spread of data around the mean. If all values are identical, there is no variation from the mean, so the standard deviation is 0. This occurs when every data point equals the mean.
Standard deviation quantifies how much individual data points deviate from the mean. If all values in a dataset are identical, each value equals the mean, so the deviation for each point is 0. Since standard deviation is the square root of the average of squared deviations, if all deviations are 0, the standard deviation is also 0. This represents the minimum possible standard deviation value.
Standard Deviation: Measure of data spread from the mean
Dispersion: How spread out values are in a dataset
Deviation: Difference between a value and the mean
• Standard deviation ≥ 0
• SD = 0 when all values are identical
• Higher SD indicates more spread
• SD of 0 means no variation
• SD is always non-negative
• SD has the same units as original data
• Thinking SD can be negative
• Confusing SD with mean
• Forgetting SD measures spread, not central tendency
Calculate the population standard deviation for the dataset: 2, 4, 6, 8, 10. Show your work.
Step 1: Calculate the mean (μ)
μ = (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6
Step 2: Calculate each squared deviation from the mean
(2-6)² = (-4)² = 16
(4-6)² = (-2)² = 4
(6-6)² = (0)² = 0
(8-6)² = (2)² = 4
(10-6)² = (4)² = 16
Step 3: Calculate the variance
σ² = (16 + 4 + 0 + 4 + 16) ÷ 5 = 40 ÷ 5 = 8
Step 4: Calculate the standard deviation
σ = √8 = 2.83
The population standard deviation is 2.83.
Calculating standard deviation involves several steps: 1) Find the mean, 2) Calculate deviations from the mean, 3) Square the deviations, 4) Find the average of squared deviations (variance), 5) Take the square root to get standard deviation. This process emphasizes that standard deviation measures the average distance from the mean, adjusted for the squaring operation.
Population Standard Deviation: When data represents entire population
Deviation: Difference between value and mean
Variance: Average of squared deviations
• Population SD: divide by N
• Always square deviations before averaging
• Take square root of variance
• Always calculate mean first
• Square deviations before averaging
• Check that SD is reasonable given data range
• Forgetting to square deviations
• Using sample formula for population data
• Not taking square root at the end
A factory produces metal rods with target length of 100cm. The lengths of 5 randomly selected rods are: 99.8cm, 100.2cm, 99.9cm, 100.1cm, 100.0cm. Calculate the sample standard deviation. If the acceptable tolerance is ±0.5cm, how consistent is the production?
Step 1: Calculate the sample mean (x̄)
x̄ = (99.8 + 100.2 + 99.9 + 100.1 + 100.0) ÷ 5 = 500.0 ÷ 5 = 100.0cm
Step 2: Calculate each squared deviation from the mean
(99.8-100.0)² = (-0.2)² = 0.04
(100.2-100.0)² = (0.2)² = 0.04
(99.9-100.0)² = (-0.1)² = 0.01
(100.1-100.0)² = (0.1)² = 0.01
(100.0-100.0)² = (0.0)² = 0.00
Step 3: Calculate the sample variance
s² = (0.04 + 0.04 + 0.01 + 0.01 + 0.00) ÷ (5-1) = 0.10 ÷ 4 = 0.025
Step 4: Calculate the sample standard deviation
s = √0.025 = 0.158cm
With SD = 0.158cm, the production is quite consistent as it's well within the ±0.5cm tolerance.
This problem demonstrates a practical application of standard deviation in quality control. Since we're analyzing a sample from the production, we use the sample standard deviation formula (dividing by n-1). The low standard deviation indicates high consistency in manufacturing, which is desirable in quality control applications.
Quality Control: Monitoring and maintaining product standards
Sample Standard Deviation: When data is a sample of population
Tolerance: Acceptable range of variation
• Sample SD: divide by (n-1)
• Low SD indicates consistency
• High SD indicates inconsistency
• Use sample SD for sample data
• Population SD for complete dataset
• Compare SD to tolerance limits
• Using population formula for sample data
• Forgetting Bessel's correction (n-1)
• Not comparing results to tolerance limits
Two students took 5 tests each. Student A scored: 80, 85, 90, 95, 100. Student B scored: 70, 80, 90, 100, 110. Calculate the standard deviation for each student's scores. Which student had more consistent performance? Explain your reasoning.
Student A:
Mean: (80+85+90+95+100)÷5 = 450÷5 = 90
Deviations: (-10)², (-5)², (0)², (5)², (10)² = 100, 25, 0, 25, 100
Variance: (100+25+0+25+100)÷5 = 250÷5 = 50
SD: √50 = 7.07
Student B:
Mean: (70+80+90+100+110)÷5 = 450÷5 = 90
Deviations: (-20)², (-10)², (0)², (10)², (20)² = 400, 100, 0, 100, 400
Variance: (400+100+0+100+400)÷5 = 1000÷5 = 200
SD: √200 = 14.14
Student A had more consistent performance with SD=7.07 compared to Student B's SD=14.14.
This example shows how standard deviation enables comparison of consistency between datasets with the same mean. Even though both students had the same average score (90), Student A's scores were more tightly clustered around the mean (lower SD), indicating more consistent performance. Student B had higher variation in scores (higher SD), showing less consistent performance.
Consistency: How similar values are to each other
Comparative Analysis: Comparing statistics between datasets
Performance Metrics: Measures of achievement or consistency
• Lower SD = more consistent
• Higher SD = more variable
• Compare SDs for same-means datasets
• Compare SDs when means are similar
• Lower SD indicates more reliability
• Use SD to assess consistency
• Comparing SDs when means are very different
• Forgetting to square deviations
• Using wrong formula (sample vs population)
Which of the following statements about standard deviation is TRUE?
The answer is B) Standard deviation is affected by outliers. Standard deviation involves squaring deviations from the mean, so extreme values (outliers) have a large impact on the calculation. Standard deviation is always non-negative, has the same units as the original data, and sample SD is typically larger than population SD due to Bessel's correction.
Standard deviation is sensitive to outliers because it involves squaring the deviations from the mean. When there are extreme values, their squared deviations contribute disproportionately to the variance, increasing the standard deviation. This sensitivity makes standard deviation a good measure of spread but also means it should be interpreted carefully when outliers are present.
Outliers: Extreme values that differ significantly from others
Sensitivity: How much a statistic changes with data changesRobustness: Resistance to influence by outliers
• SD ≥ 0 (never negative)
• SD has same units as data
• SD is sensitive to outliers
• Check for outliers before interpreting SD
• SD is always non-negative
• SD uses same units as original data
• Thinking SD can be negative
• Ignoring effect of outliers on SD
• Confusing units of SD with variance
Q: What's the difference between population and sample standard deviation?
A: The key difference is in the denominator:
The (n-1) in sample SD is called Bessel's correction and provides an unbiased estimate of population variance. Use population SD when you have data for the entire group of interest, sample SD when you have data from a subset.
Q: How is standard deviation used in real-world applications?
A: Standard deviation is widely used:
It helps quantify uncertainty and variability in all these domains.