Standard Normal Distribution Table - Stats Solver (2024)

The standard normal table, or z table, provides probabilities for the standard normal probability distribution. The standard normal probability distribution is simply a normal probability distribution with a mean of zero and a standard deviation of one. Like the normal probability distribution, the standard normal probability distribution has a bell-shape. The mean of the distribution is in the middle, which is also the highest point on the curve. Furthermore, the distribution is symmetric about the mean, with the right side of the curve being a mirror image of the left side.


Since the standard normal probability distribution is a continuous probability distribution, probabilities are given by the area under the graph. The z table gives the area under the standard normal distribution to the left of different z values. These areas are thus the probability that z will be less than or equal to that value. It is often the case that the probability you are looking for is not less than or equal but rather the probability that z will be greater than or equal or between two values. These types of probabilities involve additional steps.


The z table is made up of two pages. The first page is for negative z values and the second page is for positive z values. To find the the area (probability) to the left of a negative z-value, use the first page. For example, to find the area to the left of -1.2, match up -1.2 in the first column with .05 in the first row. The corresponding area is .1056. So that means that the probability that z will be less than or equal to -1.25 is .1056. Note that the standard normal distribution is a continuous probability distribution. That means that the probability that z will take exactly one value is zero. So the probability that z will be less than or equal to a value is the same as the probability that z will be less than that value.

z .03 .04 .05 .06 .07
-1.3 .0918 .0901 .0885 .0869 .0853
-1.2 .1093 .1075 .1056 .1038 .1020
-1.1 .1292 .1271 .1251 .1230 .1210

Calculating the probability that z will be greater than or equal to some value requires an additional step. Suppose you want to calculate the probability that z will be greater than or equal to 0.83. Start with the fact that the total area under the standard normal distribution is one. This means that the area to the right of .83 will be one minus the area to the left of .83. It is important to look at the problem in this way because the standard normal table only gives you the area to the left. Then, using the table, the area to the left of 0.83 is .7967. So the area to the right of 0.83 is 1 - .7967 = .2033.

z .01 .02 .03 .04 .05
0.7 .7611 .7642 .7673 .7704 .7734
0.8 .7910 .7939 .7967 .7995 .8023
0.9 .8186 .8212 .8238 .8264 .8289

The third type of probability to know how to calculate for the standard normal distribution is the probability that z will be between two values. For example, suppose you want to find the probability that z will be between 0.83 and 2.57. Again, the standard normal distribution only gives us the area to the left of z-values, not the area between. However, if we subtract the area to the left of the large z-value minus the area to the left of the smaller z-value, the result will be the area between them. So the area to the left of 2.57 minus the area to the left 0.83 is equal to the area between 0.83 and 2.57. Thus the probability that z will be between 0.83 and 2.57 is .9932 - .7967 = .1965.

z .05 .06 .07 .08 .09
2.4 .9929 .9931 .9932 .9934 .9936
2.5 .9946 .9948 .9949 .9951 .9952
2.6 .9960 .9961 .9962 .9963 .9964

One of the main applications of the standard normal distribution is computing probabilities for normal distributions in general. The normal distribution has many real world applications. For example, heights, weights, rainfall, test scores and many other real world phenomena follow a normal distribution. Probabilities for a normal distribution can be computed by first converting to the standard normal distribution. After conversion, the normal procedure for calculating probabilities for a standard normal distribution can be used.

Normal to Standard Normal
$ z = \dfrac{x-\mu}{\sigma} $

Aside from real-world applications, the normal distribution, and thus the standard normal distribution, is frequently used in statistical inference. The sampling distribution of the sample mean follows a normal distribution when the sample size is large. So probabilities can be calculated for the sample mean using the standard normal distribution. It is also used in confidence intervals and hypothesis testing when the population standard deviation is known. In confidence interval, the standard normal distribution is used to compute the margin of error. In hypothesis testing, the standard normal distribution is used to calculate the test statistic.

Standard Normal Distribution Table - Stats Solver (2024)

FAQs

How to calculate standard normal distribution table? ›

Step 1: Subtract the mean from the x value. Step 2: Divide the difference by the standard deviation. The z score for a value of 1380 is 1.53. That means 1380 is 1.53 standard deviations from the mean of your distribution.

What is the Z 1.96 from a table? ›

From the table, z = 1.96. Therefore 95% of the area under the standard normal distribution lies between z = -1.96 and z = 1.96.

When using a standard normal table, p(- 2 ≤ z ≤ 2 is? ›

P(−2 ≤ Z ≤ 2) = P(Z ≤ 2)−P(Z ≤ −2) = 0.9772− 0.0228 = 0.9545.

How to check z table in statistics? ›

To use a z-table, first turn your data into a normal distribution and calculate the z-score for a given value. Then, find the matching z-score on the left side of the z-table and align it with the z-score at the top of the z-table. The result gives you the probability.

How to find az score? ›

The formula for calculating a z-score is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation. Figure 2.

How do we solve for the probability using the normal table? ›

The probability of P(a < Z < b) is calculated as follows. Then express these as their respective probabilities under the standard normal distribution curve: P(Z < b) – P(Z < a) = Φ(b) – Φ(a). Therefore, P(a < Z < b) = Φ(b) – Φ(a), where a and b are positive.

What is the z value of 0.05 in the standard normal distribution table? ›

Values of the Normal distribution
area from -∞ to -z and z to ∞area from -z to zz
0.0050.9952.575829
0.010.992.326348
0.020.982.053749
0.050.951.644854
17 more rows

How to use standard normal distribution table to find p value? ›

Finding the p value from a z value, using the table with standard normal probabilities
  1. Find the row corresponding to the z value you found up to the first decimal, and the column corresponding to the second decimal. ...
  2. The two sided p value is 2×(1−pleft)

What is the formula for the z-score of a normal distribution table? ›

z = (X – μ) / σ

where X is a normal random variable, μ is the mean of X, and σ is the standard deviation of X. You can also find the normal distribution formula here.

How do you find 95% on a Z table? ›

First off, if you look at the z*-table, you see that the number you need for z* for a 95% confidence interval is 1.96. However, when you look up 1.96 on the Z-table, you get a probability of 0.975.

How to use z table to find critical value? ›

To calculate the critical z value for any confidence level, look for 1−α/2 value in the z table. For the 95% level, look for 0.975, not 0.95, to note the value of 1.96. Similarly, for 90% and 99% confidence levels, the critical z values are 1.645 and 2.575, respectively.

What is the formula for the standard normal distribution? ›

Standard Normal Distribution

f ( x ) = 1 2 π e x p ( − 1 2 x 2 ) . In other words, the standard normal distribution is the normal distribution with mean μ=0 and standard deviation σ=1 .

What is the formula for SD in normal distribution? ›

You can calculate it by subtracting each data point from the mean value and then finding the squared mean of the differenced values; this is called Variance. The square root of the variance gives you the standard deviation.

What is the formula for the normal distribution range? ›

What is the normal distribution formula? For a random variable x, with mean “μ” and standard deviation “σ”, the normal distribution formula is given by: f(x) = (1/√(2πσ2)) (e[-(x-μ)^2]/^2).

How to use az table to find critical value? ›

The z critical value can be calculated as follows:
  1. Find the alpha level.
  2. Subtract the alpha level from 1 for a two-tailed test. For a one-tailed test subtract the alpha level from 0.5.
  3. Look up the area from the z distribution table to obtain the z critical value.

References

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