Variance on TI-84 Calculator: How to Calculate Sample Variance

Variance on TI-84 Calculator, Variance is one of the most important measures of variability in statistics. It quantifies how far observations tend to spread out from the mean of a dataset.

When working with a sample, the resulting measure is called sample variance and is commonly represented by s2.

What Is Sample Variance?

Sample variance measures the average squared distance between individual observations and the sample mean.

The formula for sample variance is: s²=∑_i=1ⁿ​(xi​−xˉ)²​/n−1

where:

• s2 = sample variance
• xi​ = individual observation
• xˉ = sample mean
• n = sample size
• n−1 = degrees of freedom

The use of n−1 rather than n is known as Bessel’s correction and is used when sample variance is being used to estimate the variance of a population.

Sample Variance vs. Population Variance

It is important to distinguish between sample and population variance.

Sample variance

When your data represent a sample from a larger population: s²=∑(xi​−xˉ)^2​/n−1

Population variance

When your data represent the entire population: σ²=∑(xi​−μ)²​/N

The TI-84 reports both the sample standard deviation Sx​ and the population standard deviation σx​ in its 1-Variable Statistics output.

If you need sample variance, square Sx​.

If you need population variance, square σx​.

Example: Calculate Sample Variance

Consider the following sample:

2, 4, 4, 7, 8, 12, 14, 15, 19, 22

We will calculate the sample variance using a TI-84 calculator.

Step 1: Enter the Data

Enter the observations into the calculator:

1. Press STAT.
2. Select 1:Edit.
3. Enter the values into L1.

Your L1 list should contain:

244781214151922

Step 2: Open 1-Variable Statistics

After entering the data:

1. Press STAT.
2. Move to CALC.
3. Select 1-Var Stats.
4. Press ENTER.
5. Enter L1 if necessary.
6. Press ENTER.

The calculator will display several descriptive statistics.

Among them, you will see:

• xˉ = sample mean
• Sx​ = sample standard deviation
• σx​ = population standard deviation
• n = number of observations

For this dataset, the sample standard deviation is approximately:

Sx​≈6.783

Step 3: Calculate Sample Variance

Sample variance is simply the square of the sample standard deviation: s2=Sx2​

Therefore: s2=(6.783)2 s2≈46.01

So, the sample variance is approximately 46.01.

For greater accuracy, use the full Sx​ value displayed by the calculator before squaring rather than a rounded value.

How to Calculate Variance Directly on a TI-84

You do not need to manually calculate every deviation from the mean.

The quickest approach is:

1. Enter your data into L1.
2. Press STAT.
3. Select CALC.
4. Choose 1-Var Stats.
5. Press ENTER.
6. Locate Sx​.
7. Square the value.

The result is the sample variance.

Understanding Sx​ and σx​

One common source of confusion when using a TI-84 calculator is the difference between Sx​ and σx​.

Sx​: Sample Standard Deviation

Sx​ uses n−1 in its calculation: Sx​=n−1∑(xi​−xˉ)2​​

Squaring Sx​ gives the sample variance: Sx2​=s2

σx​: Population Standard Deviation

σx​ uses N in its calculation: σx​=N∑(xi​−μ)2​​

Squaring σx​ gives the population variance: σx2​=σ2

Therefore, do not square σx​ if you are specifically asked for sample variance.

What Does Variance Tell You?

Variance describes the amount of variability in a dataset.

A small variance means observations tend to be relatively close to the mean.

A large variance means observations tend to be more widely dispersed.

For example, consider:

Dataset A: 9, 10, 10, 11, 10

The observations are tightly clustered, so the variance is relatively small.

Compare this with:

Dataset B: 2, 6, 10, 14, 18

The observations are much more spread out, so the variance is larger.

Why Is Variance Squared?

Variance uses squared deviations from the mean: (xi​−xˉ)2

Squaring serves two important purposes:

1. It prevents positive and negative deviations from canceling each other out.
2. It gives greater weight to observations that are farther from the mean.

However, because the deviations are squared, the units of variance are also squared.

For example, if your original measurements are in meters, the variance is expressed in square meters.

This is one reason standard deviation is often easier to interpret because it is expressed in the same units as the original data.

Variance vs. Standard Deviation

Variance and standard deviation are closely related: Variance=(Standard Deviation)2

and: Standard Deviation=Variance​

For example, if: s=6.783

then: s2≈46.01

So:

• Sample standard deviation ≈ 6.783
• Sample variance ≈ 46.01

Variance vs. Range and IQR

Variance is just one way to measure dispersion.

The range uses only the minimum and maximum.

The IQR focuses on the middle 50% of observations and is relatively resistant to outliers.

Variance uses every observation and can be strongly influenced by extreme values because deviations are squared.

When Should You Use Variance?

Variance is widely used in:

• Statistical analysis
• Regression analysis
• ANOVA
• Experimental design
• Quality control
• Finance
• Machine learning
• Probability theory
• Scientific research

It is particularly important when mathematical models require a measure of variability around the mean.

Frequently Asked Questions

What is the formula for sample variance?

The formula is: s2=n−1∑(xi​−xˉ)2​

How do I calculate sample variance on a TI-84?

Enter the data into L1, select STAT → CALC → 1-Var Stats, find Sx​, and square it.

What is the difference between Sx​ and σx​?

Sx​ is the sample standard deviation, while σx​ is the population standard deviation.

Is variance the square of standard deviation?

Yes. Variance=(Standard Deviation)2

Why does sample variance use n−1?

Using n−1 degrees of freedom provides an unbiased estimator of the population variance under the usual random-sampling assumptions.

Can variance be negative?

No. Variance cannot be negative because it is based on squared deviations. It is zero only when all observations are identical.

Conclusion

Variance is a fundamental measure of variability that describes how widely observations are dispersed around their mean.

For the sample:

2, 4, 4, 7, 8, 12, 14, 15, 19, 22

the TI-84 can be used to obtain the sample standard deviation Sx​, which is approximately 6.783. Squaring the standard deviation gives a sample variance of approximately: s2≈46.01​

When using a TI-84, remember the key distinction: square Sx​ for sample variance and square σx​ for population variance. This simple distinction helps ensure that you select the correct measure for your statistical analysis.