The Central Limit Theorem states that the distribution of the sample means approaches normal regardless of the shape of the parent population.
Sample means (s) will be normally more distributed around (µ) than the individual readings (Xs). As n – the sample size – increases, then the sample averages (Xs means) will approach a normal distribution with mean (µ).
So, don’t worry if your samples are all over the place. The more sample sets you have, the sooner the averages of those sets will approach a normal distribution with a mean of (µ).
Significance of Central limit theorem
The central limit theorem is one of the most profound and useful results in all statistics and probability. The large samples (more than 30) from any sort of distribution the sample means will follow a normal distribution.
The spread of the sample means is less (narrower) than the spread of the population you’re sampling from. So, it does not matter how the original population is skewed.
- The means of the sampling distribution of the mean is equal to the population mean µx̅ =µX
- The standard deviation of the sample means equals the standard deviation of the population divided by the square root of the sample size: σ(x̅) = σ(x) / √(n)
- Samples must be independent of each other
- Samples follow random sampling
- If the population is skewed or asymmetric, the sample should have sufficiently large (for example minimum of 30 samples).
Why Central limit theorem is important
The central limit theorem allows the use of confidence intervals, hypothesis testing, DOE, regression analysis, and other analytical techniques. Many statistics have approximately normal distributions for large sample sizes, even when we are sampling from a distribution that is non-normal.
This means that we can often use well developed statistical inference procedures and probability calculations that are based on a normal distribution, even if we are sampling from a population that is not normal, provided we have a large sample size.
Mode is one of the measures of central tendency. Mode is the value that appears most often in a set of data values or a frequent number. The Unimodal distribution will have only one peak or only one frequent value in the data set. In other words, Unimodal will have only one mode, the values are increases first and reaches to peak (i.e is the mode or the local maximum) and then decreases.
Normal distribution is the best example of Unimodal. Similarly, Bimodal distribution means there are two different modes, and multimodal means more than two different modes.
Central Limit Theorem Examples
Case 1: Less than
Example: A population of 65 years male patients blood sugar was 100 mg/dL with a standard deviation is 15 mg/DL. If a sample of 4 patients’ data were drawn, what is the probability of their mean blood sugar is less than 120 mg/dL?
- µ = 100
- x̅ = 120
- σ =15
Compute P(X<120): z= x̅- µ/ σ/√n = 120-100/15/√4=20/7.5=2.66
P(x<120) = z(2.66) = 0.9961 = 99.61%
Hence, the probability of mean blood sugar is less than 120 mg/dL is 99.61%
Case 2: Between
Example: A population of 65 years male patients blood sugar was 100 mg/dL with a standard deviation is 20 mg/DL. If a sample of 9 patients’ data were drawn, what is the probability of their mean blood sugar is between 85 and 105 mg/dL?
First, compute P(x<105)
- µ = 100
- x̅ = 105
- σ =20
Compute the Z score z= x̅- µ/ σ/√n = 105-100/20/√9=5/6.67=0.75
P(x<105) = z(0.75) = 0.7734 = 77.3%
Then, compute P(x<85)
- µ = 100
- x̅ = 85
- σ =20
Compute the Z score z= x̅- µ/ σ/√n = 85-100/20/√9=-15/6.67=-2.24
P(x<85) = z(-2.24) = 0.0125 = 1.25%
Since, we are looking for blood sugar between 85 and 105 mg/dL P(85<x<105) = 77.3-1.25 = 76.05%
Hence, the probability of mean blood sugar is between 85 and 105 mg/dL is 76.05%
Case 3: Greater than
Example: A population of 65 years male patients blood sugar was 100 mg/dL with a standard deviation is 20 mg/DL. If a sample of 16 patients’ data were drawn, what is the probability of their mean blood sugar is more than 90 mg/dL?
- µ = 100
- x̅ = 90
- σ =20
Compute the Z score z= x̅- µ/ σ/√n = 90-100/20/√16=-10/5=-2
P(x<90) = z(-2.0) = 0.0228 = 2.28%
Since, we are looking for blood sugar more than 90 mg/dL =100%-2.28%=97.72%
Hence, the probability of mean blood sugar is greater than 90 mg/dL is 97.72%