Z score tables, also known as standard normal tables, are used to find the probability that a specific value, or score, will occur within a certain distribution. These tables are typically used in statistical analyses to help researchers determine the likelihood of certain events occurring based on data that has been collected.

The z score is a statistical measurement that represents the number of standard deviations a value is from the mean of a distribution. It is calculated by subtracting mean from a specific value and dividing the result by the standard deviation of the distribution.

The z score allows researchers to compare the relative position of a value within a distribution to the rest of the values in the distribution. It helps understand how unusual or rare a specific value is within a distribution.

## How to use z score table?

To use a z-score table, you must first calculate the z-score for a specific value. To do this, follow these steps:

- Determine the mean of the distribution.
- Subtract the mean from the particular value you are interested in.
- Divide the result by the standard deviation of the distribution.

Once you have calculated the z score, you can use a z score table to find the probability that this value will occur within the distribution. To do this, follow these steps:

- Find the row in the table corresponding to the whole number part of the z score.
- Find the column in the table that corresponds to the decimal part of the z score.
- The intersection of the row and column will contain the probability that the specific value you are interested in will occur within the distribution.

It is important to note that z score tables are typically only used for distributions that are approximately normally distributed. This means that the distribution is symmetrical and bell-shaped, with most values clustering around the mean. If the distribution is not normally distributed, the probabilities calculated using a z-score table may not be accurate.

## Interpreting the Results of a Z score Table

Once you have used a z score table to find the probability that a specific value will occur within a distribution, you can use this probability to interpret the results. Here are some examples of how you might interpret the results of a z score table:

- If the probability is high, the specific value is not unusual or rare within the distribution.
- If the probability is low, the specific value is unusual or rare within the distribution.
- If the probability is very low, the specific value is highly unusual or rare within the distribution.

It is important to note that the probability calculated using a z score table is the probability that the specific value will occur within the distribution. This does not necessarily mean that the value is the most likely or typical value to occur within the distribution.

## Examples of Using a Z score Table

### Example 1: Determining Probability

Consider a collection of data where the mean is 100 and the standard deviation is 15. You want to determine the probability of a value falling within 1 standard deviation of the mean.

First, you would calculate the z-score for the value you are interested in:

z = (x – μ) / σ = (100 – 100) / 15 = 0

Then, you would look up the probability for a z-score of 0 in the z-score table. You would find that the probability of a value falling within 1 standard deviation of the mean is approximately 68%.

### Example 2: Testing a Hypothesis

Suppose you want to test the hypothesis that the mean weight of a certain type of fruit is equal to 100 grams. You collect a sample of fruit and calculate the mean weight, which is 105 grams. You set the alpha level at 5%.

First, you would calculate the z-score for the sample mean:

z = (x – μ) / σ = (105 – 100) / 15 = 0.33

Then, you would look up the probability for a z-score of 0.33 in the z-score table. You would find that the probability of this sample mean occurring if the null hypothesis (that the mean weight is 100 grams) is true is approximately 37%.

Since the probability is above the alpha level of 5%, you would fail to reject the null hypothesis. This means that the sample mean of 105 grams is not statistically significant and does not provide strong evidence against the hypothesis that the mean weight is 100 grams.

### Example 3: Determining Outliers

Suppose you have a data set with a mean of 100 and a standard deviation of 15. You want to determine which values in the data set are unusual or unexpected.

First, you would calculate the z-score for each value in the data set:

z = (x – μ) / σ

Then, you would look up the probability for each z-score in the z-score table. If the probability is below 50%, it means that the value is unusual or unexpected for the data set.

For example, suppose you calculate the z-score for a value of 120 as follows:

z = (120 – 100) / 15 = 2

If you look up the probability for a z-score of 2 in the z-score table, you would find that it is approximately 2.3%. This means that the value of 120 is unusual or unexpected for the data set. It may be considered an outlier.

## Conclusion

In conclusion, a z-score table is a useful tool for interpreting the results of z-score calculationsZ-scores are measurements of how much a data point deviates from the mean of a data set, and they can be used to identify whether a value is typical or uncommon for the data set. To use a z-score table, you first need to calculate the z-score for the value you are interested in. Then, you can look up the corresponding probability in the table to interpret the results. Z-scores can be used in a variety of ways, including determining probability, testing hypotheses, and identifying outliers in data sets. Using a z-score table can help you make informed decisions and draw meaningful conclusions from your data.