GoMim Math AI | Sample Variance Calculator Guide

What is Sample Variance?

Sample variance is a key concept in statistics that measures how much the values in a sample differ from the mean. In other words, it tells us the spread or dispersion of data within a set of observations. Unlike population variance, which considers every element in an entire population, sample variance uses a smaller dataset to estimate the overall variability.

The formula for sample variance is:

$$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$$

Here,

• $s^2$ = sample variance
• $n$ = number of data points in the sample
• $x_i$ = each observation
• $\bar{x}$ = sample mean

This adjustment with $n-1$ instead of $n$ is called Bessel's correction, which helps produce an unbiased estimate of population variance.

How to Calculate Sample Variance step by step

Follow these steps to calculate sample variance manually:

1、Find the Mean (Average):

  $$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$

2、Subtract the Mean and Square the Differences:

For each data point, subtract the mean and square the result to remove negative values.

3、Add the Squared Differences:

Sum all squared differences.

4、Divide by (n – 1):

Divide the result by one less than the number of data points.

Example Calculation

Let's take a sample dataset: [4, 8, 6, 5, 3].

1、Mean:

$$\bar{x} = \frac{4 + 8 + 6 + 5 + 3}{5} = 5.2$$

2、Squared Differences:

• $(4 - 5.2)^2 = 1.44$
• $(8 - 5.2)^2 = 7.84$
• $(6 - 5.2)^2 = 0.64$
• $(5 - 5.2)^2 = 0.04$
• $(3 - 5.2)^2 = 4.84$

3、Sum:

  $$1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8$$

4、Variance:

  $$s^2 = \frac{14.8}{5-1} = 3.7$$

So, the sample variance is 3.7.

Related Practice Problem

Problem:

A company checks the weight consistency of its products. For the sample [12.1, 12.3, 11.9, 12.5, 12.2], calculate the sample variance.

Step-by-step Solution:

1、Mean:

$$\bar{x} = \frac{12.1 + 12.3 + 11.9 + 12.5 + 12.2}{5} = 12.2$$

2、Squared Differences:

• $(12.1 - 12.2)^2 = 0.01$
• $(12.3 - 12.2)^2 = 0.01$
• $(11.9 - 12.2)^2 = 0.09$
• $(12.5 - 12.2)^2 = 0.09$
• $(12.2 - 12.2)^2 = 0.00$

3、Sum:

  $$0.01 + 0.01 + 0.09 + 0.09 + 0.00 = 0.20$$

4、Variance:

  $$s^2 = \frac{0.20}{5-1} = 0.05$$

The sample variance is 0.05, indicating that the product weights are highly consistent.

Additional Practice Problem

Problem:

A teacher records the test scores of 6 students: [15, 18, 20, 22, 19, 16]. Calculate the sample variance.

Step-by-step Solution:

1、Mean:

  $$\bar{x} = \frac{15 + 18 + 20 + 22 + 19 + 16}{6} = \frac{110}{6} \approx 18.33$$

2、Squared Differences:

• $(15 - 18.33)^2 \approx 11.09$
• $(18 - 18.33)^2 \approx 0.11$
• $(20 - 18.33)^2 \approx 2.78$
• $(22 - 18.33)^2 \approx 13.44$
• $(19 - 18.33)^2 \approx 0.44$
• $(16 - 18.33)^2 \approx 5.44$

3、Sum:

  $$11.09 + 0.11 + 2.78 + 13.44 + 0.44 + 5.44 = 33.3$$

4、Variance:

  $$s^2 = \frac{33.3}{6-1} = \frac{33.3}{5} = 6.66$$

So, the sample variance is 6.66, which shows a moderate spread of test scores around the mean.

Why Sample Variance is Important

Sample variance is a fundamental statistical measure that helps quantify the variability within a dataset, providing critical insights for informed decision-making. In education, it allows teachers to evaluate differences in student performance. In engineering and manufacturing, it supports quality control and reliability testing by identifying inconsistencies. In data analysis, sample variance helps understand data distribution, detect outliers, and improve predictions. It is also essential in hypothesis testing and constructing confidence intervals, making it a vital tool for researchers, analysts, and professionals. Using a sample variance calculator can streamline these analyses and ensure accurate results

Use GoMim Math AI Solver for Sample Variance

The GoMim Math AI Solver acts as a powerful sample variance calculator, simplifying calculations and saving time. By entering your dataset into GoMim's AI math solver, you can instantly compute sample variance with step-by-step explanations, eliminating manual errors. This allows students, researchers, and analysts to focus on interpreting results rather than performing complex arithmetic. Whether for homework, research, or professional data analysis, GoMim's calculator ensures fast, precise, and reliable solutions. Try it now to experience how efficiently sample variance problems can be solved.

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FAQ

Q: What is the difference between sample variance and population variance?

A: Sample variance is calculated from a subset of a larger population, and it uses n-1 in the denominator to correct for bias. Population variance uses the entire data set and divides by n.

Q: Why do we use n-1 in the sample variance formula?

A: Using n-1 rather than n corrects for the bias in estimating a population parameter from a sample, providing a more accurate representation.

Q: Can sample variance be negative?

A: No, sample variance cannot be negative because it is calculated from squared differences, which are always non-negative.

Q: How does sample variance relate to standard deviation?

A: Sample variance is the square of the sample standard deviation. Standard deviation is often preferred for interpretation as it is in the same units as the data.

Q: Is sample variance the same as variance?

A: Sample variance refers specifically to variance calculated from a sample of data, while variance can also refer to population variance.

Q: How do AI tools assist in calculating sample variance?

A: AI tools like GoMim provide quick and accurate computations, reducing human error and saving time in analyzing data sets.