Normal Distribution Percentile

Normal Distribution Percentile Meaning

A normal distribution is a probability distribution where the data is distributed about the mean symmetrically to look like a bell-shaped curve, which is sometimes called a density curve.

Normal distributions are generally more suitable for large data sets. Many naturally occurring data, like test scores or organisms’ mass, tend to pattern themselves close to a normal distribution.

The normal distribution curve shown in the graph below, shows that the majority of the data is clustered around the middle of the graph, right where the mean is located.

The graph then tapers off towards the left and the right ends, to show smaller portion of the data far from the mean. Half of the data falls below the mean, and half of the data falls above the mean and thus, the mean is also the median of the data. The highest point on the graph is located at the middle of the graph as well, therefore this is where the mode is.

So, for a normal distribution, the mean, median, and mode are all equal.

Furthermore, the curve is divided into pieces by the standard deviations. The area under the normal distribution curve represents 100% of the data. For a standard normal distribution, this means that the area under the curve is equal to 1.

A specific percentage of the data is assigned to each standard deviation away from the mean on a normal distribution. These specific percentages are called the Empirical Rule of Normal Distribution,

• About 68% of the data falls within 1 standard deviation of the mean.
• About 95% of the data falls within 2 standard deviations of the mean.
• About 99.7% (almost all of teh data!) falls within 3 standard deviations of the mean.

This is sometimes called the "68-95-99.7 Rule".

Standard Normal Distribution with standard deviation percentages.

Those percentages are very helpful in knowing information about the repartition of the data. But one of the most important pieces of information to know about a data value in a normal distribution, is how much of the data it is greater than or less than a specific value, called the percentile.

Normal Distribution Percentile of Mean

As stated earlier in the above paragraph, the mean in the normal distribution curve lies right in its middle. The curve distributes thus the data symmetrically about the mean, that is 50% of the data are above the mean and 50% of the data are below the mean. This means that the mean is the 50th percentile of the data.

We take the following example to understand this better.

Does the Standard deviation has a percentile of its own as well? Let's figure this out in the next paragraph!

Normal Distribution Percentile of Standard Deviation

A very good question that one could have is the following, what is the percentile for each standard deviation?

Well, knowing that the mean is the 50th percentile, and recalling what does each percentage represent in every section of the normal distribution graph, you can figure out the percentile at each standard deviation.

For 1 standard deviation above the mean, that is to the right of the mean, find the percentile by adding the 34.13% above the mean to the 50% to get 84.13%. Usually for percentile, you round to the nearest whole number.

So, 1 standard deviation is about the 84th percentile.

If you wanted to find the percentile of 2 standard deviations, you would continue to add the percentages to the right of the mean to 50%. Therefore, the second standard deviation's percentile is 13.59% and 34.13% added to 50%, that gives you 97.72%, or about the 98th percentile.

And thus, 2 standard deviations are about the 98% percentile.

For finding the percentile of a standard deviation below the mean, that is to the left of the mean, subtract the standard deviation's percentage from 50%.

For 1 standard deviation below the mean, find the percentile by subtracting 34.13% from 50% to get 15.87%, or about the 16th percentile.

You can subtract the next standard deviation percentage to find the percentile of 2 standard deviations below the mean, 15.87% - 13.59% is 2.28%, or about the 2nd percentile.

The following normal distribution graph shows the corresponding percentage that lie below each standard deviation.

Fig. 1. Standard normal distribution showing the percentage of data below each standard deviation.

Normal Distribution Percentile Formula

When working with a normal distribution, you will not just be interested in the percentile of the standard deviations, or the mean's percentile. In fact, sometimes you will work with values that fall somewhere between the standard deviations, or you may be interested in a specific percentile that does not correspond to one of the standard deviations mentioned above, nor the mean.

And this is where the need of a normal distribution percentile formula arises. In order to do so, we recall the following definition of z-score.

The importance of the z-score is that not only it tells you about the value itself, but where it is located on the distribution.

Conversely, in order to find a value based on a given percentile, the z-score formula can be reformulated into \[x=\mu+Z\sigma.\]

Luckily, you probably won't have to calculate the percentile every time for the z-score you want, that would be rather burdensome! Instead, you can use a z-score table, like the ones below.

A z-score table has the proportion of the data that falls below each z-score so that you can find the percentile directly.

Fig. 2. Negative z-score table for a normal distribution

Fig. 3. Positive z-score table for a normal distribution.

How to read a z-score table in order to find the percentile?

Once you have found your z-score, follow these steps for using the z-score to find the corresponding percentile. Most z-score tables show z-scores out to the hundredths place, but you can find more precise tables if needed.

Reading a z-score table can be done using the following steps,

Step 1. Look at the z-score you are given or have found.

Step 2. Look along the left side of the table, which shows the ones and the tenths places of your z-score. Find the row that matches your first two digits.

Step 3. Look along the top of the table, which shows the hundredths place. Find the column that matches your third digit.

Step 4. Find the intersection of the row and the column that matches your ones, tenths, and hundredths places. This is the proportion of data below your z-score, which is equal to the percentage of data below your z-score.

Step 5. Multiply by 100 to get a percentage. Generally, you round to the nearest whole number to get a percentile.

Normal Distribution Percentile Graph

The graph below shows a standard normal distribution curve with a few common percentiles marked with their corresponding z-scores.

Fig. 4. Standard normal distribution with z-scores for common percentiles.

Notice that these percentiles are symmetric, just like the standard deviations. The 25th percentile and the 75th percentile are both 25 percentile points away from the mean, so their z-scores are both 0.675, with the only difference being the negative to show that the 25th percentile is below the mean. The same is true for the 10th and 90th percentiles.

This can be helpful when you want to find percentiles that may be presented differently.

Let's say that someone were to report that they scored in the top 10th percentile of a test. That obviously sounds very good, but the 10th percentile is well below the mean, right? Well, they are not really saying that they are in the tenth percentile. They are indicating that they scored lower than only 10% of the other test-takers. This is equivalent to saying they scored higher than 90% of the test-takers, or rather scored in the 90th percentile.

Knowing that normal distribution is symmetric allows flexibility in how we view the data.

The graphs above and the z-score tables all are based on the standard normal distribution that has a mean of 0 and a standard deviation of 1. This is used as the standard so that it is scalable for any data set.

But, obviously, most data sets do not have a mean of zero or a standard deviation of 1. That is what the z-score formulas can help with.

Examples of Normal Distribution Percentile

Growth charts, test scores, and probability problems are common problems you will see when working with normal distributions.

You may also need to find a value based on a certain percentile. For the most part, that will involve doing the steps above in reverse.

Normal Distribution Proportion

Normal distributions are so useful because they are proportional to each other via the z-score and percentiles.

Each normal distribution may have its own mean and standard deviation, which can affect the spread of the data. But the proportion of the data that lies within each standard deviation is the same across all normal distributions. Each area under the curve represents a proportion of the data set or the population.

This means that you can find the percentile for any value in any normal distribution as long as you know the mean and standard deviation.

Let's look at the two following examples of standardized tests to compare.

Comparing Data Using Normal Distribution

Because all normal distributions are proportional, you are able to compare the data from two different sets, with different means and standard deviations, as long as both are normally distributed.