Standard Normal Distribution

Updated on Oct 02, 2024

Curve & Z-Table

There are a lot of shapes and distributions in statistics, but the most common is the normal distribution. A normal distribution is symmetric. The standard normal curve allows us to look at how the symmetry works to the left and the right of the middle mean. We can analyze any amount of standard deviations (also called Z-scores) to the left or right, in order to see exactly how the data is spreading from the center.

The Standard Normal Curve

The normal distribution has a few interesting characteristics

It is bell curve (a symmetric hill shape)
It is symmetric, matching equally on the left and right
The mean is in the center of the symmetric “bell” distribution (where I have placed the orange arrow below), and the standard deviation is the distance from that mean (each yellow double arrow is pointing to the distance of one standard deviation) to the next point along the curve

The picture below shows an example of a general bell curve with a mean of zero and a standard deviation of one

Standard Normal Dist Curve & Z-Table.pdf - Google Drive - Google Chrome

Z-Scores

A Z-Score is a point along the standard normal distribution.

Z implies a standard normal distribution
A critical value, Z score, or specified number of standard deviations shows the starting point of an area under the curve, to the left or right of that point.
The area under the curve is also called a proportion under the curve, percentage, or probability

Technology Help: Z-scores

In StatCrunch

Select Stat, then Calculators, then Normal

When using the normal distribution calculator, you can find the area to the left and right of a point The point can be noted as Z, or the number of standard deviations

Less than (<) indicates to the left, and Greater than (>) indicates to the right
When you enter the point you want to consider, you will find the approximate area to the left or right of that point, the probability under the normal curve

*Note: Leave the mean & standard deviation alone at this point. This will change as you encounter real world problems!

In StatKey

Under theoretical distributions, Select the “Normal Distribution” option
Check off the tail direction you want
Leave the mean and standard deviation alone at this point. This will change as you encounter real world problems!

In Excel

1. Utilize the standard normal distribution function in Excel “Norm.S...” to find area under the curve, the same way as in StatCrunch

Excel always calculates to the left of the number, or less than the Zpoint!

2. To find a point greater than Z, or to the right of Z, you must subtract from 100% or 1

Below, you can see the functions to solve for the probability under the curve/area under the curve in Excel

3. Be sure to enter your specific Z value.

Utilizing the Z-score table

*Remember that different table values may vary slightly so read them thoroughly.

The Z-table is similar to Excel. It will show you the area under the curve. However, it splits the number into two parts. The beginning of the number is in the first column in blue. The end of the number is in the top row in yellow

For instance: If I wanted to find the area under the curve for -1.26. I would need to first split the number into: -1.2 and 0.06. If I follow the blue column down to -1.2 and move across to 0.06, I will then find the probability/area to the LEFT of -1.26, highlighted in teal. Find the area to the right of that value by subtracting from 1, just as in Excel

Z	0.09	0.08	0.07	0.06	0.05	0.04	0.03	0.02	0.01	0.0
-3	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.0013
-2.9	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.0019
-2.8	0.001	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.0026
-2.7	0.002	0.002	0.002	0.002	0.003	0.003	0.003	0.003	0.003	0.0035
-2.6	0.003	0.003	0.003	0.003	0.004	0.004	0.004	0.004	0.004	0.0047
-2.5	0.004	0.004	0.005	0.005	0.005	0.005	0.005	0.005	0.006	0.0062
-2.4	0.006	0.006	0.006	0.006	0.007	0.007	0.007	0.007	0.008	0.0082
-2.3	0.008	0.008	0.008	0.009	0.009	0.009	0.009	0.010	0.010	0.0107
-2.2	0.011	0.011	0.011	0.011	0.012	0.012	0.012	0.013	0.013	0.0139
-2.1	0.014	0.014	0.015	0.015	0.015	0.016	0.016	0.017	0.017	0.0179
-2	0.018	0.018	0.019	0.019	0.020	0.020	0.021	0.021	0.022	0.0228
-1.9	0.023	0.023	0.024	0.025	0.025	0.026	0.026	0.027	0.028	0.0287
-1.8	0.029	0.030	0.030	0.031	0.032	0.032	0.033	0.034	0.035	0.0359
-1.7	0.036	0.037	0.038	0.039	0.040	0.040	0.041	0.042	0.043	0.0446
-1.6	0.045	0.046	0.047	0.048	0.049	0.050	0.051	0.052	0.053	0.0548
-1.5	0.055	0.057	0.058	0.059	0.060	0.061	0.063	0.064	0.065	0.0668
-1.4	0.068	0.069	0.070	0.072	0.073	0.074	0.076	0.077	0.079	0.0808
-1.3	0.082	0.083	0.085	0.086	0.088	0.090	0.091	0.093	0.095	0.0968
-1.2	0.098	0.100	0.102	0.103	0.105	0.107	0.109	0.111	0.113	0.1151
-1.1	0.117	0.119	0.121	0.123	0.125	0.127	0.129	0.131	0.133	0.1357
-1	0.137	0.140	0.142	0.144	0.146	0.149	0.151	0.153	0.156	0.1587
-0.9	0.161	0.163	0.166	0.168	0.171	0.173	0.176	0.178	0.181	0.1841
-0.8	0.186	0.189	0.192	0.194	0.197	0.200	0.203	0.206	0.209	0.2119
-0.7	0.214	0.217	0.220	0.223	0.226	0.229	0.232	0.235	0.238	0.242
-0.6	0.245	0.248	0.251	0.254	0.257	0.261	0.264	0.267	0.270	0.2743
-0.5	0.277	0.281	0.284	0.287	0.291	0.294	0.298	0.301	0.305	0.3085
-0.4	0.312	0.315	0.319	0.322	0.326	0.33	0.333	0.337	0.340	0.3446
-0.3	0.348	0.352	0.355	0.359	0.363	0.366	0.370	0.374	0.378	0.3821
-0.2	0.382	0.389	0.393	0.397	0.401	0.405	0.409	0.412	0.416	0.4207
-0.1	0.424	0.428	0.432	0.436	0.440	0.444	0.448	0.452	0.456	0.4602

Now, you try! Practice Problems

Example One

What is the mean in the image below? What is the standard deviation?

Example Two

What is the area under the curve to the left of Z = -2.1? You can use any method and obtain the same solution. Feel free to try with Excel, StatCrunch, Statkey, the Z-score chart, or ALL.

Example One

Example Solution One

The mean is in the middle of this symmetric ‘bell’ normal model at 122. The standard deviation is the distance of one ‘step’ away from the center,3. The average is 122. The deviation from the average is 3

Example Solution Two

The area under the curve when Z < -2.1 is 0.0179, rounded to 4 decimal places. The probability is 0.0179. In 100 random trials, the chance of a deviation of -2.1 is approximately 100*0.0179 or 1.79 out of 100, or 1.79%. See the technology results below:

With Z-Table

Z
	…	…	…	…	…	…	…	…	…	…
	0.003	0.003	0.003	0.003	0.004	0.004	0.004	0.004	0.004	0.004
	0.004	0.004	0.005	0.005	0.0054	0.005	0.005	0.005	0.006	0.006
	0.006	0.006	0.006	0.006	0.0071	0.007	0.007	0.007	0.008	0.008
	0.008	0.008	0.008	0.009	0.0094	0.009	0.009	0.010	0.010	0.010
	0.011	0.011	0.011	0.011	0.0122	0.012	0.012	0.013	0.013	0.013
	0.014	0.014	0.015	0.015	0.0158	0.016	0.016	0.017	0.017
	0.018	0.018	0.019	0.019	0.0202	0.020	0.021	0.021	0.022	0.022
	0.023	0.023	0.024	0.025	0.0256	0.026	0.026	0.027	0.028	0.028
	…	…	…	…	…	…	…	…	…	…

With StatCrunch

*Note: The same result is found with each technology, though Statkey does round the value.

With Excel

With StatKey

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