6.4: Using Tables of the Normal Distribution

Z-scores III

Do z-score probabilities always need to be calculated as the chance of a value either above or below a given score? How would you calculate the probability of a z-score between -0.08 and +1.92?

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Z-Scores

To calculate the probability of getting a value with a z-score between two other z-scores, you can either use a reference table to look up the value for both scores and subtract them to find the difference, or you can use technology. In this lesson, which is an extension of Z-scores and Z-scores II, we will practice both methods.

Historically, it has been very common to use a z-score probability table like the one below to look up the probability associated with a given z-score:

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	Z
0.0	.5	0.504	0.508	0.512	0.516	0.5199	0.5239	0.5279	0.5319	0.5359	0.0
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753	0.1
0.2	0.5793	0.5832	0.5871	0.591	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141	0.2
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.648	0.6517	0.3
0.4	.6554	0.6591	0.6628	0.6664	0.67	0.6736	0.6772	0.6808	0.6844	0.6879	0.4
0.5	0.6915	0.695	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.719	0.7224	0.5
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549	0.6
0.7	0.758	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852	0.7
0.8	0.7881	0.791	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133	0.8
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.834	0.8365	0.8389	0.9
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621	1.0
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.877	0.879	0.881	0.883	1.1
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.898	0.8997	0.9015	1.2
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177	1.3
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319	1.4
1.5	0.9332	0.9345	0.9357	0.937	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441	1.5
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545	1.6
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633	1.7
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706	1.8
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.975	0.9756	0.9761	0.9767	1.9
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817	2.0
2.1	0.9821	0.9826	0.983	0.9834	0.9838	0.9842	0.9846	0.985	0.9854	0.9857	2.1
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.989	2.2
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916	2.3
2.4	0.9918	0.992	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936	2.4
2.5	0.9938	0.994	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952	2.5
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.996	0.9961	0.9962	0.9963	0.9964	2.6
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.997	0.9971	0.9972	0.9973	0.9974	2.7
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.998	0.9981	2.8
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986	2.9
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.999	0.999	3.0
3.1	0.999	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993	3.1
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995	3.2
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997	3.3
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998	3.4
3.5	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	3.5
3.6	0.9998	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	3.6
3.7	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	3.7
3.8	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	3.8
3.9	1	1	1	1	1	1	1	1	1	1	3.9
Z	0.000	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	Z

Since the proliferation of the Internet, however, you can also use a free online calculator.

Calculating Probability

What is the probability associated with a z-score between 1.2 and 2.31?

To evaluate the probability of a value occurring within a given range, you need to find the probability of both the upper and lower values in the range, and subtract to find the difference.

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• First find z=1.2 on the z-score probability reference above: .8849 Remember that value represents the percentage of values below 1.2.
• Next, find and record the value associated with z=2.31: .9896
• Since approximately 88.49% of all values are below z=1.2 and approximately 98.96% of all values are below z=2.31, there are 98.96%−88.49%=10.47% of values between.

Calculating the Probability of a Value Occurring in a Normal Distribution

1. What is the probability that a value with a z-score between -1.32 and +1.49 will occur in a normal distribution?

Let’s use the online calculator on "Math Portal" for this one.

When you open the page, you should see a window like this:

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All you need to do is select the radio button to the left of the first type of probability, input “-1.32” into the first box, and 1.49 into the second. When you click “Compute”, you should get the result

P(−1.32<Z<1.49)=0.8385

Which tells us that there is approximately and 83.85% probability that a value with a z-score between 1.32 and 1.49 will occur in a normal distribution.

Notice that the calculator also details the steps involved with finding the answer:

1. Estimate the probability using a graph, so you have an idea of what your answer should be.
2. Find the probability of z < 1.49, using a reference. (0.9319)
3. Find the probability of z < −1.32, again, using a reference. (0.0934)
4. Subtract the values: 0.9319−0.0934=0.8385 or 83.85%

2. What is the probability that a random selection will be between 8.45 and 10.25, if it is from a normal distribution with μ=10 and σ=2?

This question requires us to first find the z-scores for the value 8.45 and 10.25, then calculate the percentage of value between them by using values from a z-score reference and finding the difference.

1. Find the z-score for 8.45, using the z-score formula: ^(x−μ)/_σ

^8.45−10/₂=^−1.55/₂≈−0.78

2. Find the z-score for 10.25 the same way:

^10.25−10/₂=^0.25/₂≈.13

3. Now find the percentages for each, using a reference (don’t forget we want the probability of values less than our negative score and less than our positive score, so we can find the values between):

P(Z<−0.78)=.2177 or 21.77%

P(Z<.13)=.5517 or 55.17%

4. At this point, let’s sketch the graph to get an idea what we are looking for:

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5. Finally, subtract the values to find the difference:

.5517−.2177=.3340 or about 33.4%

There is approximately a 33.4% probability that a value between 8.45 and 10.25 would result from a random selection of a normal distribution with mean 10 and standard deviation 2.

Earlier Problem Revisited

Do z-score probabilities always need to be calculated as the chance of a value either above or below a given score? How would you calculate the probability of a z-score between -0.08 and +1.92?

After this lesson, you should know without question that z-score probabilities do not need to assume only probabilities above or below a given value, the probability between values can also be calculated.

The probability of a z-score below -0.08 is 46.81%, and the probability of a z-score below 1.92 is 97.26%, so the probability between them is 97.26%−46.81%=50.45%.

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Examples

1. What is the probability of a z-score between -0.93 and 2.11?
2. What is P(1.39<Z<2.03)?
3. What is P(−2.11<Z<2.11)?

Solutions:

Example 1

What is the probability of a z-score between -0.93 and 2.11

Using the z-score probability table above, we can see that the probability of a value below -0.93 is .1762, and the probability of a value below 2.11 is .9826. Therefore, the probability of a value between them is .9826−.1762=.8064 or 80.64%

Example 2

What is P(1.39<Z<2.03)?

Using the z-score probability table, we see that the probability of a value below z=1.39 is .9177, and a value below z=2.03 is .9788. That means that the probability of a value between them is .9788−.9177=.0611 or 6.11%

Example 3

What is P(−2.11<Z<2.11)?

Using the online calculator on "Math Portal", we select the top calculation with the associated radio button to the left of it, enter “-2.11” in the first box, and “2.11” in the second box. Click “Compute” to get “.9652”, and convert to a percentage. The probability of a z-score between -2.11 and +2.11 is about 96.52%.

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Review

Find the probabilities, use the table from the lesson or an online resource.

1. What is the probability of a z-score between +1.99 and +2.02?
2. What is the probability of a z-score between -1.99 and +2.02?
3. What is the probability of a z-score between -1.20 and -1.97?
4. What is the probability of a z-score between +2.33 and-0.97?
5. What is the probability of a z-score greater than +0.09?
6. What is the probability of a z-score greater than -0.02?
7. What is P(1.42<Z<2.01)?
8. What is P(1.77<Z<2.22)?
9. What is P(−2.33<Z<−1.19)?
10. What is P(−3.01<Z<−0.71)?
11. What is P(2.66<Z<3.71)?
12. What is the probability of the random occurrence of a value between 56 and 61 from a normally distributed population with mean 62 and standard deviation 4.5?
13. What is the probability of a value between 301 and 329, assuming a normally distributed set with mean 290 and standard deviation 32?
14. What is the probability of getting a value between 1.2 and 2.3 from the random output of a normally distributed set with μ=2.6 and σ=.9?

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Review (Answers)

To view the Review answers, open this PDF file and look for section 9.5.

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Vocabulary

Term	Definition
z-score	The z -score of a value is the number of standard deviations between the value and the mean of the set.
z-score table	A z-score table associates the various common z-scores between 0 and 3.99 with the decimal probability of being less than or equal to that z-score.

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Additional Resources

Practice: Using Tables of the Normal Distribution

Real World: No Country for Old Men