How to find sample variance on ti 84

How to find sample variance on ti 84, The TI-84 graphing calculator can handle many common statistical calculations, including variance, standard deviation, and critical values used in hypothesis testing and confidence intervals.

These functions are useful for students, researchers, and data analysts who need to perform statistical calculations without calculating every value manually. The steps depend on whether your data represent an entire population or a sample and which statistical distribution your analysis requires.

How to Calculate Variance on a TI-84 Calculator

Variance measures how far observations spread around their mean. A small variance indicates that the values tend to be close to the mean, while a larger variance indicates greater variability.

The TI-84 calculates standard deviation directly. You can square the appropriate standard deviation to obtain variance.

Step 1: Enter Your Data

1. Press STAT.
2. Select 1:Edit.
3. Enter your observations into list L1.

For example, enter these five values:

4, 6, 8, 10, 12

Step 2: Calculate the Statistics

1. Press STAT.
2. Use the right arrow to select CALC.
3. Choose 1:1-Var Stats.
4. Enter L1 if it is not already selected.
5. Press ENTER.

The calculator displays several statistics, including the mean, sample standard deviation, and population standard deviation.

Look for:

• Sx: Sample standard deviation.
• σx: Population standard deviation.
• x̄: Sample mean.

Step 3: Calculate the Variance

Square the appropriate standard deviation.

• Sample variance: (s^2=(Sx)^2)
• Population variance: (\sigma^2=(\sigma x)^2)

For the data (4,6,8,10,12), the mean is 8.

The sum of the squared deviations from the mean is:

[
(4-8)^2+(6-8)^2+(8-8)^2+(10-8)^2+(12-8)^2=40
]

The sample variance is:

[
s^2=\frac{40}{5-1}=10
]

The population variance is:

[
\sigma^2=\frac{40}{5}=8
]

Use sample variance when your observations represent a sample from a larger population. Use population variance when the data include the entire population of interest.

How to Find a Critical Value on a TI-84

A critical value defines a boundary for a statistical test or confidence interval. Its value depends on the significance level, whether the test is one-tailed or two-tailed, and the distribution being used.

The TI-84 supports several inverse distribution functions, including invNorm( and invT(.

Finding a Z Critical Value with invNorm

A z critical value is used when a standard normal distribution is appropriate, such as in many large-sample procedures when the population standard deviation is known.

For a 95% two-sided confidence interval, the significance level is:

[
\alpha=1-0.95=0.05
]

Each tail contains (0.025) of the probability, so the cumulative probability to the left of the positive critical value is (0.975).

On the TI-84:

1. Press 2nd.
2. Press VARS to open the DISTR menu.
3. Select invNorm(.
4. Enter 0.975.
5. Press ENTER.

The result is approximately:

[
z^*=1.96
]

For a 95% two-sided confidence interval, the critical values are approximately (-1.96) and (1.96).

For a one-tailed test with a significance level of 0.05, use a cumulative probability of 0.95 for an upper-tail critical value, giving approximately (1.645).

Finding a t Critical Value with invT

The t distribution is commonly used for inference about a population mean when the population standard deviation is unknown, particularly with smaller samples under appropriate assumptions.

For a 95% two-sided confidence interval with 10 degrees of freedom:

[
\alpha=0.05,\qquad df=10
]

Each tail has a probability of (0.025). The cumulative probability for the positive critical value is therefore (0.975).

On the TI-84:

1. Press 2nd, then VARS.
2. Select invT( from the distribution menu.
3. Enter 0.975,10.
4. Press ENTER.

The result is approximately:

[
t^*=2.228
]

For this example, the critical values are approximately (-2.228) and (2.228).

The second argument specifies the degrees of freedom. For a one-sample t procedure, this is usually the sample size minus one.

Which TI-84 Function Should You Use?

The inverse functions return critical values from cumulative probabilities. The cumulative distribution functions calculate probabilities between specified bounds.

Common Mistakes When Using a TI-84

Confusing sample and population variance: Squaring Sx gives sample variance, while squaring σx gives population variance. Choosing the wrong value changes the result.

Entering the wrong cumulative probability: For a 95% two-sided interval, use 0.975 to find the positive critical value, not 0.95.

Using the wrong distribution: A z critical value and a t critical value are not interchangeable. Choose the distribution that matches the statistical procedure.

Ignoring degrees of freedom: The t critical value changes with the degrees of freedom. Always enter the correct value for the test or interval.

Using the wrong tail probability: For a one-tailed test at (\alpha=0.05), an upper-tail critical value uses cumulative probability 0.95. For a two-tailed test at the same significance level, the positive critical value uses 0.975.

Conclusion

The TI-84 simplifies variance calculations and critical-value lookups for common statistical problems. Use 1-Var Stats to obtain the appropriate standard deviation, square it to calculate variance, and use invNorm( or invT( to find critical values. Correctly identifying the population or sample, distribution, significance level, and degrees of freedom is essential for obtaining reliable results.