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Central limit theorem is defined as the mean value of all samples of a given population being equal to the mean of the population in approximate measures given that the sample size of the population is fairly large and has a finite variation. The most important aspect of the central limit theorem is that the average of the sample means and standard deviations will equal the population mean and standard deviation. A large sample size can predict the characteristics of the given population more accurately.
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Key terms: Standard Deviation, Central Limit Theorem Probability, Central Limit Theorem, Sample Mean, Population Mean
Central Limit Theorem Definition
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Central Limit Theorem is defined as the overall distribution of a given sample mean, is approximately the same as the normal distribution given condition, the sample size gets bigger and we assume that all the samples are similar to each other, irrespective of the shape of the total population distribution.
Central Limit Theorem
Central Limit Theorem Statistics Example
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Central Limit Theorem can be explained with the help of the following example:
Assume that you have 10 different debate teams in your school and each team consists of 100 students. Now, we need to find out the average marks of all these students across all the teams. How will we do it when there are so many teams and so many students?
Well, the easiest way in which we can find the average height of all students is by determining the average of their marks. To do so, we will first need to determine the marks of each student and then add them all.
Then, we will need to divide the total sum of their marks by the total number of the students and we will get the average marks of the students. While calculating the average through this method, we will first choose the students randomly from different teams and design a sample. Every sample will consist of 20 students. Then, we will follow the steps mentioned below:
- First, all the samples will be taken, and the mean of each sample will be determined individually. Then, we will determine the mean of these sample means.
- Through this process we will get the approximate mean marks of all the students who are a part of the debate team.
- Now, if we will find the histogram of the mean marks sample then we can observe a bell-shaped curve.
Note: It is very important to take samples that are large enough in size. When we take samples that are larger in size, it means that the sample mean distribution is becoming normal when we calculate it by repeated sampling.
Central Limit Theorem Formula
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As we have learned how to define the central limit theorem and have also seen the examples. Now let us learn and understand the formula of the Central Limit Theorem.
We can apply the Central Limit Theorem for a larger sample size, i.e., when n ≥ 30.
The formula of the Central Limit Theorem are as follows:
\(\mu^-_x = \mu\)and
\(\sigma_x^- = \frac{\mu}{\sqrt{n}}\)Where,
μ = Population mean
σ = Population standard deviation
\(\mu^-_x\)= Sample mean \(\sigma_x^-\)= Sample standard deviationn = Sample size
Things to Remember
- Central Limit Theorem is the overall distribution of a given sample mean.
- With the help of the central limit theorem, we can calculate the mean efficiently.
- Sample standard deviation can be calculated by dividing standard deviation with the square root of the sample size.
- Sample mean is equal to the population mean.
- In central limit theorem, the average of the sample means and standard deviations will equal the population mean and standard deviation.
Sample Questions
Ques. State the central limit theorem. (2 Marks)
Ans. The Central Limit Theorem states that the distribution of all the samples is approximately equal to the normal distribution when the sample size gets larger, given the condition that the samples taken are all similar in size, irrespective of the shape of the population distribution.
Ques. What are the applications of the central theorem in statistics? (3 Marks)
Ans. The different applications of the Central Theorem are as follows:
- If the distribution is not normal or is unknown, we take into consideration that the sample distribution is normal according to the Central Limit Theorem. This method tends to assume that the given population is distributed normally. This, in turn, helps us to analyze the data in methods such as building confidence intervals.
- For estimating the mean of the population more accurately, we tend to increase the samples that are taken from the population which would ultimately decrease the mean deviation of the samples.
- For creating the range of different values that are likely to have the population mean, we can make use of the sample mean.
Ques. A certain group of welfare recipients receives SNAP benefits of $110 per week with a standard deviation of $20. If a random sample of 25 people is taken, what is the probability their mean benefit will be greater than $120 per week? (2 Marks)
Ans. Step 1: Insert the information into the z-formula:
= (120-110)/20 √25 = 10/ (20/5) = 10/4 = 2.5.
Step 2: Look up the z-score in a table (or calculate it using technology). A z-score of 2.5 has an area of roughly 49.38%. Adding 50% (for the left half of the curve), we get 99.38%.
Ques. A population of 29-year-old males has a mean salary of $29,321 with a standard deviation of $2,120. If a sample of 100 men is taken, what is the probability their mean salaries will be less than $29,000? (3 Marks)
Ans. Step 1: Insert the values into the z-formula:
= (29,000 – 29,321) / (2,120/√100) = -321/212 = -1.51.
Step 2: Look up the z-score in the left-hand-z-table (or use technology). -1.51 has an area of 93.45%.
However, this is not the answer, as the question is asking for LESS THAN, and 93.45% is the area “greater than” so you need to subtract from 100%.
100% – 93.45% = 6.55% or about 0.07.
Ques. There are 250 dogs at a dog show who weigh an average of 12 pounds, with a standard deviation of 8 pounds. If 4 dogs are chosen at random, what is the probability they have an average weight of greater than 8 pounds and less than 25 pounds? (5 Marks)
Ans. Step 1: Identify the parts of the problem. Your question should state:
- mean (average or μ) standard deviation (σ)population size
- Sample size (n)
- number associated with “less than” 1
- number associated with “greater than” 2
Step 2: Draw a graph. Label the center with the mean. Shade the area between 1 and 2. This step is optional, but it may help you see what you are looking for.
Step 3: Use the following formula to find the z-scores.
\(z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}}\)All this formula is asking you to do is:
a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 25 – 12 = 13.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8 / √ 4 = 4
c) Divide your result from a by your result from b: 13 / 4 = 3.25
Step 4: Use the formula from Step 3 to find the z-values. This time, use Xbar2 from Step 1 (8).
a) Subtract the mean (μ in Step 1) from the greater than value (Xbar in Step 1): 8 – 12 = -4.
b) Divide the standard deviation (σ in Step 1) by the square root of your sample (n in Step 1): 8 / √ 4 = 4
c) Divide your result from a by your result from b: -4 / 4= -1
Step 5: Look up the value you calculated in Step 3 in the z-table.
Z value of 3.25 corresponds to .4994
Step 6: Look up the value you calculated in Step 4 in the z-table.
Z value of 1 corresponds to .3413
Note that the bell curve is symmetrical, so if you want to look up a negative value like -1, then just look up the positive counterpart. The area will be the same.
Step 7: Add Step 5 and 6 together:
.4994 + .3413 = .8407
Step 8: Convert the decimal in Step 7 to a percentage:
.8407 = 84.07%
Ques. A fertilizer company manufactures organic fertilizer in 10-pound bags with a standard deviation of 1.25 pounds per bag. What is the probability that a random sample of 15 bags will have a mean between 9 and 9.5 pounds? (2 Marks)
Ans. Step 1: 2nd VARS 2.
Step 2: Enter your variables (lower bound, upper bound, mean, and standard deviation). Separate each variable by a comma: 9,9.5, 10,(1.25/√15)).
Step 3: Press ENTER. This returns the probability of .05969, or .05969%.
Ques. What is a linear function? (1 Marks)
Ans. A linear function is a relation between two variables that produces a straight line when graphed.
Ques. Define Slope. (2 Marks)
Ans. Slope is a measure of the steepness of a line. A line can have a positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the over the change in the .” The symbol for slope is .
Ques. Define standard deviation. (2 Marks)
Ans. A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
Ques. What do you mean by the central limit theorem? (2 Marks)
Ans. The Central Limit Theorem is defined as the overall distribution of a given sample mean, is approximately the same as the normal distribution given condition, the sample size gets bigger and we assume that all the samples are similar to each other, irrespective of the shape of the total population distribution.
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