# The Normal Distribution Table Definition

## What Is the Normal Distribution?

The normal distribution formula is based on two basic parameters that characterize the features of a particular dataset: mean and standard deviation.

The mean represents the “centre” or average value of the whole dataset, while the standard deviation represents the “spread” or variety of data points around that mean value.

### Key Takeaways

- The normal distribution formula is based on two basic parameters that characterize the features of a particular dataset: mean and standard deviation.
- The standard conversion to Z-values, which form part of the Normal Distribution Table, was created to promote an unified standard procedure for straightforward computations and application to real-world issues.
- The normal curve is symmetrical around the mean; the mean is in the center and splits the area in half; the entire area under the curve equals 1 for mean=0 and stdev=1; and the distribution is fully defined by its mean and stdev.
- In securities trading, normal distribution tables are used to assist detect uptrends and downtrends, support and resistance levels, and other technical indicators.

## Normal Distribution Example

Consider the following 2 datasets:

- Dataset 1 = 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10
- 6, 8, 10, 12, 14, 14, 12, 10, 8, 6 = Dataset 2

Mean = 10 and standard deviation (stddev) = 0 for Dataset1.

Mean = 10 and standard deviation (stddev) = 2.83 for Dataset2.

Let’s plot these values for DataSet1:

Similarly for DataSet2:

In both graphs above, the red horizontal line represents the “mean” or average value of each dataset (10 in both cases).In the second graph, the pink arrows represent the dispersion or variance of data values from the mean value. In the example of DataSet2, this is represented as a standard deviation of 2.83. Because DataSet1 has all values the same (as 10 each) and no variances, the standard deviation is zero, thus no pink arrows are relevant.

The stddev number has a few important and beneficial properties that are particularly useful in data analysis. The data values in a normal distribution are symmetrically distributed on each side of the mean. The following graph is created for any normally distributed dataset by producing a graph with stddev on the horizontal axis and the number of data values on the vertical axis.

## Properties of a Normal Distribution

- The normal curve is symmetrical with respect to the mean.
- The mean is at the center and splits the region in half.
- For mean=0 and stdev=1, the entire area under the curve is equal to one.
- The mean and standard deviation of the distribution fully characterize it.

As seen in the graph above, stddev indicates the following:

- Within one standard deviation of the mean (-1 to +1), 68.3% of data values are within one standard deviation of the mean.
- 95.4% of data points are within two standard deviations of the mean (-2 to +2).
- 99.7% of data points are within three standard deviations of the mean (-3 to +3).

When measured, the area under the bell-shaped curve represents the desired probability of a certain range:

- less than X: For example, the chance of data values being less than 70.
- more than X: For example, the likelihood of data values being higher than 95.
- between X1 and X2: for example, the chance of data values ranging between 65 and 85.

where X is a value of interest (examples below).

Plotting and calculating the area is not always easy since various datasets have varied mean and standard deviation values. The standard conversion to Z-values, which form part of the Normal Distribution Table, was created to promote a unified standard procedure for straightforward computations and application to real-world issues.

Z = (X – mean)/stddev, where X is the random variable.

Essentially, this conversion requires the mean and standard deviation to be normalized to 0 and 1, respectively, allowing a predefined set of Z-values (from the Normal Distribution Table) to be utilized for simple computations. The following is a snapshot of a conventional z-value table including probability values:

z

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.0

0.00000

0.00399

0.00798

0.01197

0.01595

0.01994

…

0.1

0.0398

0.04380

0.04776

0.05172

0.05567

0.05966

…

0.2

0.0793

0.08317

0.08706

0.09095

0.09483

0.09871

…

0.3

0.11791

0.12172

0.12552

0.12930

0.13307

0.13683

…

0.4

0.15542

0.15910

0.16276

0.16640

0.17003

0.17364

…

0.5

0.19146

0.19497

0.19847

0.20194

0.20540

0.20884

…

0.6

0.22575

0.22907

0.23237

0.23565

0.23891

0.24215

…

0.7

0.25804

0.26115

0.26424

0.26730

0.27035

0.27337

…

…

…

…

…

…

…

…

…

To get the probability associated with a z-value of 0.239865, first round it to two decimal places (i.e. 0.24).Then, in the rows, look for the first two significant digits (0.2) and in the column, look for the least significant digit (the remaining 0.04). This results in a value of 0.09483.

The entire normal distribution table may be seen here, with accuracy up to 5 decimal points for probability values (including those for negative values).

Let’s look at some real-world instances. Individuals’ heights in a big group follow a normal distribution pattern. Assume we have a group of 100 people whose heights are recorded, and the mean and standard deviation are estimated to be 66 and 6 inches, respectively.

Here are a few examples of queries that a z-value table may readily answer:

### What is the probability that a person in the group is 70 inches or less?

The question is to determine the cumulative value of P(X=70), that is, how many values in the total dataset of 100 will be between 0 and 70.

Let’s start by converting the X-value of 70 to the appropriate Z-value.

Z = (X – mean) / standard deviation = (70-66)/6 = 4/6 = 0.66667 = 0.67 (round to 2 decimal places)

We must now calculate P (Z = 0.67) = 0. 24857. (from the z-table above)

That is, there is a 24.857% chance that someone in the group is shorter than or equal to 70 inches.

But wait—the preceding is insufficient. Remember, we’re seeking for the probability of all potential heights between 0 and 70. The fraction from mean to target value is just shown above (i.e. 66 to 70).To get the proper result, we must add the other half (from 0 to 66).

Because 0 to 66 represents half of the range (one extreme to the mid-way mean), the probability is just 0.5.

As a result, the accurate chance of a person being 70 inches or shorter = 0.24857 + 0.5 = 0. 74857 = 74.857%

These are the two summed zones that depict the answer graphically (by computing the area):

### What is the probability that a person is 75 inches or higher?

i.e. Find Complementary cumulativeP(X>=75).

Z = (X – mean)/stddev = (75-66)/6 = 9/6 = 1.5

P (Z >=1.5) = 1- P (Z <= 1.5) = 1 – (0.5+0.43319) = 0.06681 = 6.681%

### What is the probability of a person being in between 52 inches and 67 inches?

Find P(52<=X<=67).

P(52<=X<=67) = P [(52-66)/6 <= Z <= (67-66)/6] = P(-2.33 <= Z <= 0.17)

= P(Z <= 0.17) –P(Z <= -0.233) = (0.5+0.56749) - (.40905) =

This normal distribution table (together with z-values) is often used in stock market probability estimates for stocks and indexes. They are used in range-based trading to detect uptrends and downtrends, support and resistance levels, and other technical indicators based on the mean and standard deviation of a normal distribution.

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