## What is the Central Limit Theorem?

The central limit theorem is a statistical theorem that states that, for a sufficiently large sample size, the distribution of the mean of any random variable will be approximately normal, regardless of the underlying distribution.

One of the key assumptions of the central limit theorem is that the sample size must be sufficiently large. In general, the larger the sample size, the more likely it is that the distribution of the mean will be normal. For example, if we have a sample of 100 observations, we would expect the distribution of the mean to be more normal than if we only had a sample of 10 observations.

The central limit theorem is important because it allows us to use normal distributions to approximate any underlying distribution. This is useful because many statistical techniques are based on normality (e.g., t-tests and ANOVA). Therefore, if we can assume that our data are approximately normally distributed, we can use these statistical techniques even if the underlying distribution is not actually normal.

## What is the z-score?

In statistics, the z-score (or standard score) of an observation is the number of standard deviations that observation is above or below the mean. For example, if the z-score is 2.5, this means that the observation is 2.5 standard deviations above the mean.

The z-score is also known as the standard score, and it is a measure of how many standard deviations an observation is from the mean. The z-score formula for central limit theorem states that for a normally distributed population with a mean of μ and a standard deviation of σ, the z-score of an observation x is:

z = (x – μ) / σ

where x is the value of the observation, μ is the population mean, and σ is the population standard deviation.

## How to calculate the z-score

The z-score is a numerical measurement that indicates how many standard deviations an element is from the mean of a data set. A z-score of 2.5, for example, would indicate that the element is 2.5 standard deviations from the mean. The z-score is also known as the standard score.

### Step 1: Find the mean

To calculate the z-score, you first need to find the mean. This is simply the average of all the numbers in your data set. To find the mean, add up all the numbers and divide by the number of items in your data set.

For example, let’s say you have five test scores:

87

93

77

67

73

The mean would be (87 + 93 + 77 + 67 + 73) / 5 = 738 / 5 = 147.6.

The mean is always represented by the letter μ (mu).

### Step 2: Subtract the mean from each data point

Now that you have your raw data set, it’s time to calculate the z-scores. This step is pretty simple. For each data point, you just need to subtract the mean from that data point. So, if X is your data point and μ is the mean:

X – μ = z-score

For example, let’s say you want to calculate the z-score for 114. To do this, you would take 114 and subtract the mean from it. So, if the mean was 100, your calculation would look like this:

114 – 100 = 14

This means that 114 is 14 standard deviations away from the mean.

### Step 3: Square each of the differences

After you have the mean and standard deviation of the population, and you know how many items are in your sample, you can calculate the z-score.

To calculate the z-score, take each x value (data point) in your sample, subtract the population mean $\mu$ from it, and then divide that difference by the population standard deviation $\sigma$:

$$z=\frac{x-\mu}{\sigma}$$

This formula gives you the z-score for each x value. To get the z-scores for all of the values in your sample, simply calculate this formula for each x value.

### Step 4: Add up all of the squared differences

Now that we have the mean and standard deviation for our distribution of X, we can calculate the z-score for a given X value. The z-score tells us how many standard deviations above or below the mean our X value is.

We’ll use the following formula to calculate the z-score:

z = (X – μ) / σ

where:

z is the z-score

X is our data point of interest

μ is the mean of our data set

σ is the standard deviation of our data set

### Step 5: Divide the sum by the number of data points

After finding the sum of all the data points, you need to divide it by the number of data points that you have. This will give you the mean of the data set. In order to find the z-score, you need to take the difference of each data point and the mean, and then divide it by the standard deviation.

## How to use the z-score to find the probability of an event

The z-score is a mathematical formula used to calculate the number of standard deviations away from the mean of a dataset. The z-score can be used to find the probability of an event occurring. The z-score is also known as the standard score.

### Step 1: Convert the z-score to a p-value

The z-score is a numerical measurement that describes a value’s relationship to the mean of a data set. It is calculated by subtracting the data set’s mean from a individual value, and then dividing that difference by the data set’s standard deviation.

A z-score can be placed on a bell curve realize where it lies in relation to the rest of the data set. The mean of the bell curve is always 0, and values less than 0 are located below the mean while values above are located above the mean. The further a z-score is from 0, indiciating it is more extreme, the rarer it will be.

You can convert a z-score to a p-value to find out the probability that an event will occur. To do this, you need to know where on the bell curve your z-score falls. You can use a Z-table (a table of all possible z-scores) to find this information.

Once you have converted your z-score to a p-value, you can interpret it to find out how likely it is for an event to happen. A low p-value (usually <= 0.05) indicates that an event is unlikely to happen by chance and may be significant. A high p-value (> 0.05) indicates that an event is likely to happen by chance and is not significant

### Step 2: Use the p-value to find the probability of an event

Now that you know how to calculate the z-score for a given situation, you can use a z-score table (found in the back of most statistics textbooks) to find the corresponding p-value. Once you have the p-value, you can immediately find the probability of an event occurring by using either of the following two methods:

1) Convert the p-value to a percentile.

2) Find the percentage of area under the normal curve that is to the left of (or right of) your z-score.