Central Limit Theorem (CLT) on Calculator
The Central Limit Theorem (CLT) is one of the most important concepts in statistics. It explains why the sampling distribution of the sample mean can often be approximated by a normal distribution, even when the original population is not normally distributed.
This makes the CLT extremely useful for calculating probabilities, constructing confidence intervals, and performing statistical inference.
What Is the Central Limit Theorem?
The Central Limit Theorem states that, under appropriate conditions, the sampling distribution of the sample mean becomes approximately normal as the sample size increases, regardless of the shape of the underlying population distribution.
If a population has mean μ and standard deviation σ, and a random sample of size n is selected, then the sampling distribution of the sample mean has:
Mean of the Sampling Distribution
The mean of the sampling distribution of the sample mean is equal to the population mean:
μx̄ = μ
Standard Deviation of the Sampling Distribution
The standard deviation of the sampling distribution is called the standard error:
σx̄ = σ / √n
Therefore, for sufficiently large samples:
x̄ ≈ N(μ, σ/√n)
The larger the sample size, the smaller the standard error becomes. This means that sample means tend to cluster more closely around the population mean as the sample size increases.
When Can You Use the Central Limit Theorem?
The CLT is particularly useful when:
- The population distribution is not normal.
- A sufficiently large random sample is available.
- Observations are independent.
- The sample is representative of the population.
There is no universal sample size that guarantees normality. A sample size of 30 is often used as a rough rule of thumb, but the appropriate size depends on the shape of the population distribution. Highly skewed or heavy-tailed populations may require larger samples.
Using a TI-84 Calculator for CLT Problems
The TI-84 calculator’s normalcdf( function can be used to calculate probabilities involving the sampling distribution of a sample mean.
The general syntax is:
normalcdf(lower, upper, mean, standard deviation)
For a sampling distribution of the mean, use:
normalcdf(lower, upper, μ, σ/√n)
Where:
- lower = lower boundary of the sample mean
- upper = upper boundary of the sample mean
- μ = population mean
- σ/√n = standard error of the sample mean
To access normalcdf( on a TI-84:
- Press 2nd.
- Press VARS to open the DISTR menu.
- Select normalcdf(.
Example 1: Probability Between Two Values
Suppose a population has a mean of 70 and a standard deviation of 7. A random sample of 35 observations is selected.
What is the probability that the sample mean is between 68 and 72?
First calculate the standard error:
SE = σ / √n
SE = 7 / √35 ≈ 1.183
The probability can then be calculated using:
normalcdf(68,72,70,7/√35)
The result is approximately:
P(68 < x̄ < 72) ≈ 0.909
Therefore, there is approximately a 90.9% probability that the sample mean will fall between 68 and 72.
Example 2: Probability Greater Than a Value
Suppose a population has a mean of 50 and a standard deviation of 4. A random sample of 30 observations is selected.
What is the probability that the sample mean is greater than 48?
First calculate the standard error:
SE = 4 / √30 ≈ 0.730
On the TI-84, enter:
normalcdf(48,1E99,50,4/√30)
The result is approximately:
0.9969
Therefore, the probability that the sample mean is greater than 48 is approximately 99.69%.
TI-84 Tip
You can use a very large positive number such as 1E99 as the upper boundary when calculating a right-tail probability.
On the TI-84, 1E99 can be entered using the calculator’s scientific notation function.
Example 3: Probability Less Than a Value
Suppose a population has a mean of 20 and a standard deviation of 3. A random sample of 40 observations is selected.
What is the probability that the sample mean is less than 19?
Calculate the standard error:
SE = 3 / √40 ≈ 0.474
Enter the following into the TI-84:
normalcdf(-1E99,19,20,3/√40)
The result is approximately:
0.0175
Therefore, there is approximately a 1.75% probability that the sample mean will be less than 19.
How Sample Size Affects the Sampling Distribution
One of the most important applications of the CLT is understanding how sample size affects the standard error.
The standard error is:
SE = σ / √n
For example, suppose σ = 10.
| Sample Size | Standard Error |
|---|---|
| 10 | 3.162 |
| 25 | 2.000 |
| 50 | 1.414 |
| 100 | 1.000 |
| 400 | 0.500 |
As the sample size increases, the standard error decreases.
This means larger samples generally produce sample means that are more concentrated around the population mean.
CLT vs. Population Distribution
It is important to distinguish between the population distribution and the sampling distribution of the sample mean.
The population itself might be:
- Normal
- Skewed
- Uniform
- Irregular
- Heavily skewed
The CLT concerns what happens to the distribution of sample means when repeated random samples are taken.
For example, even if individual observations come from a strongly skewed population, the distribution of sample means can become approximately normal when the sample size is sufficiently large.
Common Mistakes When Using the CLT
Using σ Instead of σ/√n
When calculating probabilities for a sample mean, don’t use the population standard deviation directly.
Incorrect:
normalcdf(68,72,70,7)
Correct:
normalcdf(68,72,70,7/√35)
The standard deviation of the sampling distribution is the standard error, not the population standard deviation.
Confusing x With x̄
The CLT deals with the distribution of the sample mean, written as x̄.
The standard error is:
σx̄ = σ/√n
Assuming n = 30 Always Works
A sample size of 30 is commonly mentioned as a guideline, but it is not a universal requirement. The required sample size depends on the population distribution and how close an approximation is needed.
Central Limit Theorem Formula
The standardized form of the sampling distribution can be written as:
Z = (x̄ − μ) / (σ/√n)
This formula converts a sample mean into a z-score and can then be used to calculate probabilities from the standard normal distribution.
Frequently Asked Questions
What does the Central Limit Theorem tell us?
The CLT tells us that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes sufficiently large, under appropriate assumptions.
Does the original population have to be normal?
No. The CLT is especially useful when the underlying population is not normally distributed.
What happens to the standard error when sample size increases?
The standard error decreases according to:
SE = σ/√n
Therefore, larger samples generally produce more precise estimates of the population mean.
What TI-84 function is used for CLT probability calculations?
The normalcdf( function is commonly used when the sampling distribution of the mean can be modeled using a normal distribution.
Conclusion
The Central Limit Theorem provides the foundation for many statistical procedures involving sample means. By understanding the relationship between the population mean, population standard deviation, sample size, and standard error, you can calculate probabilities for sample means even when the underlying population is not normally distributed.
The TI-84 calculator makes these calculations convenient through the normalcdf( function. The key step is remembering that the standard deviation used for the sampling distribution of the mean is the standard error, σ/√n, rather than σ itself.
Once you understand this distinction, you can use the TI-84 to solve a wide range of CLT probability problems quickly and accurately.
