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📝 Standard Deviation Examples

Example 1: Student Test Scores

Data Set: 85, 90, 78, 92, 88, 76, 95, 82

Step 1: Calculate the mean = (85 + 90 + 78 + 92 + 88 + 76 + 95 + 82) / 8 = 686 / 8 = 85.75

Step 2: Find the deviations: (85-85.75)=-0.75, (90-85.75)=4.25, (78-85.75)=-7.75, (92-85.75)=6.25, (88-85.75)=2.25, (76-85.75)=-9.75, (95-85.75)=9.25, (82-85.75)=-3.75

Step 3: Square the deviations: 0.5625, 18.0625, 60.0625, 39.0625, 5.0625, 95.0625, 85.5625, 14.0625

Step 4: Sum of squared deviations = 317.5

Population variance (σ²): 317.5 / 8 = 39.69

Population std dev (σ): √39.69 = 6.30

Sample variance (s²): 317.5 / 7 = 45.36

Sample std dev (s): √45.36 = 6.73

The test scores have a sample standard deviation of approximately 6.73 points from the mean.

Example 2: Daily Temperatures (°C)

Data Set: 22, 25, 19, 23, 27, 21, 24

Step 1: Calculate the mean = (22 + 25 + 19 + 23 + 27 + 21 + 24) / 7 = 161 / 7 = 23

Step 2: Deviations: -1, 2, -4, 0, 4, -2, 1

Step 3: Squared deviations: 1, 4, 16, 0, 16, 4, 1

Step 4: Sum of squared deviations = 42

Population variance (σ²): 42 / 7 = 6.00

Population std dev (σ): √6.00 = 2.45

Sample variance (s²): 42 / 6 = 7.00

Sample std dev (s): √7.00 = 2.65

Daily temperatures vary from the mean by about 2.65°C (sample std dev).

Example 3: Heights (cm)

Data Set: 165, 170, 172, 168, 175, 180, 162, 178, 169, 171

Step 1: Calculate the mean = 1710 / 10 = 171

Step 2: Deviations: -6, -1, 1, -3, 4, 9, -9, 7, -2, 0

Step 3: Squared deviations: 36, 1, 1, 9, 16, 81, 81, 49, 4, 0

Step 4: Sum of squared deviations = 278

Population variance (σ²): 278 / 10 = 27.80

Population std dev (σ): √27.80 = 5.27

Sample variance (s²): 278 / 9 = 30.89

Sample std dev (s): √30.89 = 5.56

The heights have a sample standard deviation of 5.56 cm from the average height of 171 cm.

📖 Formulas & Guide

Population Standard Deviation: σ = √( Σ(xᵢ - μ)² / N )

Where μ is the population mean and N is the total number of values

Sample Standard Deviation: s = √( Σ(xᵢ - x̄)² / (n - 1) )

Where x̄ is the sample mean and n is the sample size (Bessel's correction)

Population Variance: σ² = Σ(xᵢ - μ)² / N

The average of the squared differences from the population mean

Sample Variance: s² = Σ(xᵢ - x̄)² / (n - 1)

Uses n-1 in denominator for unbiased estimation (Bessel's correction)

How to Calculate Standard Deviation

Find the mean (average) — Add all values and divide by the count.
Calculate deviations — Subtract the mean from each value.
Square each deviation — Eliminates negative values.
Sum the squared deviations — Add all squared values together.
Divide by N (population) or n-1 (sample) — This gives you the variance.
Take the square root — This gives you the standard deviation.

Population vs. Sample: When working with data from an entire population (every member), divide by N. When working with a sample (a subset), divide by n-1 to get an unbiased estimate of the population standard deviation. The sample formula with n-1 is called Bessel's correction.

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❓ Frequently Asked Questions

What is the difference between population and sample standard deviation?

Population standard deviation (σ) is used when you have data for every member of the entire group you're studying. It divides by N (the total number of values).

Sample standard deviation (s) is used when you have a subset of the population. It divides by n-1 (Bessel's correction) to provide an unbiased estimate of the population standard deviation. You should use the sample version in most real-world scenarios since you rarely have access to an entire population.

Why do we use n-1 instead of n for sample standard deviation?

Using n-1 (Bessel's correction) corrects the bias in the estimation of the population variance and standard deviation from a sample. A sample tends to be less variable than the full population it came from. Dividing by n would systematically underestimate the true population variability. By dividing by n-1 instead of n, we get a larger (and less biased) estimate that better approximates the true population standard deviation.

What does a low vs. high standard deviation tell me?

A low standard deviation means the data points tend to be clustered closely around the mean (average). The data is more consistent and predictable.

A high standard deviation means the data points are spread out over a wider range of values. There is more variability and less consistency.

For example, test scores with s = 5 show more consistency than scores with s = 20. In finance, a low standard deviation in stock returns indicates lower volatility and risk.

Can standard deviation be negative?

No, standard deviation can never be negative. Standard deviation is the square root of variance, which is calculated from squared deviations (always non-negative). Since we take the square root of a non-negative number, the result is always zero or positive. A standard deviation of zero means all values in the data set are identical.

What is the relationship between variance and standard deviation?

Variance is the square of the standard deviation, and standard deviation is the square root of variance. Both measure the spread of data around the mean. Variance is expressed in squared units (e.g., cm² for height data), while standard deviation is expressed in the original units (e.g., cm), making it more interpretable. This is why standard deviation is more commonly used in practice — it's easier to relate to the actual data values.

When should I use population vs. sample standard deviation in real life?

Use population standard deviation when you have data for every single member of the group (e.g., test scores of all students in one class, all countries in the UN).

Use sample standard deviation when you have a subset of the population (e.g., a survey of 500 voters out of millions, measuring heights of 50 randomly selected people from a city). Most real-world research and data analysis uses the sample standard deviation because complete population data is rarely available.

⚠️ Important Note: Standard deviation is sensitive to outliers — a single extreme value can significantly increase the standard deviation. Always check your data for outliers and consider using the median alongside the mean for a more complete picture of your data distribution. For normally distributed data, approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.

📊 Standard Deviation Calculator

📋 Step-by-Step Calculation

📝 Standard Deviation Examples

Example 1: Student Test Scores

Example 2: Daily Temperatures (°C)

Example 3: Heights (cm)

📖 Formulas & Guide

How to Calculate Standard Deviation

❓ Frequently Asked Questions

📊 Standard Deviation Calculator

📋 Step-by-Step Calculation

📝 Standard Deviation Examples

Example 1: Student Test Scores

Example 2: Daily Temperatures (°C)

Example 3: Heights (cm)

📖 Formulas & Guide

How to Calculate Standard Deviation

❓ Frequently Asked Questions

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