To learn what the sampling distribution of (overlineX) is once the sample size is large. To learn what the sampling distribution of (overlineX) is when the populace is normal.

You are watching: As the size of the sample increases, what happens to the shape of the distribution of sample means?

In Example 6.1.1, we built the probability distribution of the sample intend for samples of dimension 2 attracted from the population of four rowers. The probcapability distribution is:

<eginarrayc arx & 152 & 154 & 156 & 158 & 160 & 162 & 164\ hline P(arx) &dfrac116 &dfrac216 &dfrac316 &dfrac416 &dfrac316 &dfrac216 &dfrac116\ endarray>

Figure (PageIndex1) shows a side-by-side compariboy of a histogram for the original population and also a histogram for this circulation. Whereas the distribution of the populace is unicreate, the sampling circulation of the mean has actually a shape approaching the form of the acquainted bell curve. This phenomenon of the sampling circulation of the intend taking on a bell form also though the population distribution is not bell-shaped happens in general. Here is a rather more realistic example. Figure (PageIndex1): Distribution of a Population and also a Sample Mean

Suppose we take samples of size (1), (5), (10), or (20) from a populace that consists entirely of the numbers (0) and (1), half the population (0), half (1), so that the populace expect is (0.5). The sampling distributions are:

(n = 1):

<eginarrayc arx & 0 & 1 \ hline P(arx) &0.5 &0.5 \ endarray onumber>

(n = 5):

<eginarrayc arx & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \ hline P(arx) &0.03 &0.16 &0.31 &0.31 &0.16 &0.03 \ endarray onumber>

(n = 10):

<eginarrayc c c c c c c c c c c arx & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \ hline P(arx) &0.00 &0.01 &0.04 &0.12 &0.21 &0.25 &0.21 &0.12 &0.04 &0.01 &0.00 \ endarray onumber>

(n = 20):

<eginarrayc c c c c c c c c c c arx & 0 & 0.05 & 0.10 & 0.15 & 0.20 & 0.25 & 0.30 & 0.35 & 0.40 & 0.45 & 0.50 \ hline P(arx) &0.00 &0.00 &0.00 &0.00 &0.00 &0.01 &0.04 &0.07 &0.12 &0.16 &0.18 \ endarray onumber>

and

<eginarrayc c c c c c c c c c arx & 0.55 & 0.60 & 0.65 & 0.70 & 0.75 & 0.80 & 0.85 & 0.90 & 0.95 & 1 \ hline P(arx) &0.16 &0.12 &0.07 &0.04 &0.01 &0.00 &0.00 &0.00 &0.00 &0.00 \ endarray onumber>

Histograms showing these distributions are presented in Figure (PageIndex2). Figure (PageIndex2): Distributions of the Sample Mean

As (n) rises the sampling distribution of (overlineX) evolves in an amazing way: the probabilities on the reduced and the top ends shrink and the probabilities in the middle end up being larger in relation to them. If we were to proceed to rise (n) then the form of the sampling circulation would certainly end up being smoother and also more bell-shaped.

What we are seeing in these examples does not depfinish on the particular population distributions associated. In basic, one might begin through any type of circulation and also the sampling circulation of the sample expect will progressively resemble the bell-shaped normal curve as the sample dimension rises. This is the content of the Central Limit Theorem.

## The Central Limit Theorem

For samples of size (30) or even more, the sample mean is around commonly distributed, through suppose (mu _overlineX=mu) and also typical deviation (sigma _overlineX=dfracsigma sqrtn), wbelow (n) is the sample size. The bigger the sample dimension, the much better the approximation. The Central Limit Theorem is shown for several common populace distributions in Figure (PageIndex3). Figure (PageIndex3): Distribution of Populations and Sample Means

The damelted vertical lines in the numbers situate the population mean. Regardless of the circulation of the populace, as the sample size is boosted the shape of the sampling circulation of the sample suppose becomes increasingly bell-shaped, focused on the population suppose. Generally by the time the sample size is (30) the circulation of the sample suppose is practically the very same as a normal circulation.

The importance of the Central Limit Theorem is that it enables us to make probcapability statements around the sample suppose, specifically in relation to its value in comparikid to the populace expect, as we will certainly view in the examples. But to use the result appropriately we have to initially realize that tbelow are 2 sepaprice random variables (and therefore 2 probcapability distributions) at play:

(X), the measurement of a single aspect schosen at random from the population; the distribution of (X) is the distribution of the populace, with intend the populace intend (mu) and also typical deviation the populace traditional deviation (sigma); (overlineX), the suppose of the measurements in a sample of size (n); the distribution of (overlineX) is its sampling distribution, via mean (mu _overlineX=mu) and traditional deviation (sigma _overlineX=dfracsigma sqrtn).

For samples of any dimension attracted from a normally spread population, the sample mean is generally distributed, with mean (μ_X=μ) and standard deviation (σ_X =σ/sqrtn), where (n) is the sample dimension.

The effect of raising the sample dimension is presented in Figure (PageIndex4). Figure (PageIndex4): Distribution of Sample Means for a Normal Population

Example (PageIndex3)

A protokind automotive tire has a architecture life of (38,500) miles through a typical deviation of (2,500) miles. Five such tires are made and tested. On the presumption that the actual population expect is (38,500) miles and also the actual population traditional deviation is (2,500) miles, discover the probability that the sample suppose will be much less than (36,000) miles. Assume that the circulation of lifetimes of such tires is normal.

Solution:

For simplicity we usage systems of hundreds of miles. Then the sample suppose (overlineX) has actually intend (mu _overlineX=mu =38.5) and typical deviation (sigma _overlineX=dfracsigma sqrtn=dfrac2.5sqrt5=1.11803). Due to the fact that the populace is usually distributed, so is (overlineX), hence

<eginalign* P(overlineXExample (PageIndex4)

An automobile battery manufacturer clintends that its midgrade battery has a expect life of (50) months with a conventional deviation of (6) months. Suppose the distribution of battery lives of this particular brand is approximately normal.

On the assumption that the manufacturer’s clintends are true, discover the probcapacity that a randomly schosen battery of this form will certainly last less than (48) months. On the exact same presumption, uncover the probcapacity that the suppose of a random sample of (36) such batteries will be less than (48) months.

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Solution:

Because the population is known to have a normal distribution

<eginalign* P(X The sample suppose has mean (mu _overlineX=mu =50) and conventional deviation (sigma _overlineX=dfracsigma sqrtn=dfrac6sqrt36=1). Thus

<eginalign* P(overlineX

## Key Takeaway

When the sample size is at least (30) the sample suppose is normally dispersed. When the population is normal the sample mean is generally dispersed regardless of the sample dimension.