Normal distributions do not necessarily have the same means and standard deviations. Each data set or distribution of scores will have their own mean, standard deviation and shape - even when they follow a normal distribution.
A normal distribution with a mean of 0 (u=0) and a standard deviation of 1 (o= 1) is known a standard normal distribution or a Z-distribution.
Z-scores describe the exact location of every score in a distribution, for example:
What the distance between the score and the mean is in terms of the number of standard deviations. For example: a Z-score of 1.0 is one standard deviation above the mean; a Z-score of -2.5 is two and a half standard deviations below the mean. A Z-score of 0 will represent the mean of the distribution.
A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the formula:
Where x is the standardised value- or value on the standard normal distribution, x is the value on the original distribution, µ is the mean of the original distribution, and o is the standard deviation of the original distribution.
To calculate a Z-score, the mean and standard deviation are needed.
For example, if the mean of a normal distribution of class test scores is 50, and the standard deviation is 10, to calculate the Z-score for 26 the formula is applied:
Z = 26-50 = -2.4
10
The Z-score of a raw score of 26, in this given distribution, is -2.4 (negative sign means that the score is below the mean).
Transforming a Z-score back into raw score
Additionally, a Z-score can be transformed into a raw score by:
In the example above, the raw score -2.4 can be transformed back into a raw score by:
10,000 students have taken a survey to measure perceived stress. The mean is 36, and standard deviation is 8. With this information, calculate:
Q1- The standardised - Z- score for a student who obtained 36 (You have attempts left)
Your response is correct. The Z score of the mean value is always 0.
Your response is incorrect, please try again. Note that the mean of the distribution is 36.
Your response is incorrect. The correct answer is 0.
10,000 students have taken a survey to measure perceived stress. The mean is 36, and standard deviation is 8. With this information, calculate:
Q2- The standardised - Z- score for a student who obtained 44 (You have attempts left)
Your response is correct. The formula looks like this 44-36/8= 1.
Your response is incorrect, please try again.
The correct answer is 44-36/8= 1.
10,000 students have taken a survey to measure perceived stress. The mean is 36, and standard deviation is 8. With this information, calculate:
Q3- The standardised - Z- score for a student who obtained 28 (You have attempts left)
Your response is correct. The formula looks like this 28-36/8= -1.
Your response is incorrect, please try again..
The correct answer is 28-36/8= -1.
The standard normal curve table is used to calculate the precise percentage of scores between the mean (Z-score of 0) and any other Z-score. It can be used to determine:
The standard normal curve table is used:
Calculating percentage of scores above or below a Z-score
Summary of Steps for determining percentage above or below Z-score:
There are different versions of the standard normal curve table. In this version, the Z column contains values of the standard normal distribution; the second column contains the area below Z. Since the distribution has a mean of 0 and a standard deviation of 1, the Z column is equal to the number of standard deviations below (or above) the mean. For example:
How to use a standard normal curve table
Once the scores of a distribution have been converted into standard or Z-scores, a normal distribution table can be used to calculate percentages and probabilities. Since the normal distribution is a continuous distribution, the probability that X is greater than or less than a particular value can be found.
A normal curve table gives the precise percentage of scores between the mean (Z-score = 0) and any other Z score. The normal curve table can be used to:
The table gives the proportion to the left of a chosen Z-value of up to 2 decimal places. To read the table, find the Z score in the left column Z. If your Z score contains decimals, use the columns to the right. For example, if you are looking for a Z score of 0.75, you will look at the intersection of 0.7 (Z column) and the column 0.05 (0.7 + 0.05= 0.75).
To obtain the probabilities, simply multiply the percentage by 100. E.g. 0.7734 would be expressed as 77.34%.
Example 1: Finding the percentage of values to the left of a Z score
In a standard normal distribution, what percentage of values will be less than 1.28?
1. Draw a diagram: you are looking for the percentage of the graph to the left of 1.28.
2. Use the standard normal table to find the value to the left of 1.28.
3. The value is .89973, which means that the percentage of values less than 1.28 is 89.97%
Example 2: Finding the percentage of values to the right of a Z-value
In a standard normal distribution, what percentage of values will be above 1.28?
1. Draw a diagram: in this example, you are looking for the percentage of values to the right of 1.28.
2. As the table only gives us values to the left of a Z score, we will use the percentage of values to the left of 1.28 that we found in the previous example:
Example 3: Finding the percentage of values between the mean and a particular Z-score
What percentage of values are between 0 and 1.28?
1. First draw a diagram: in this case, you are looking for values between the mean (0) and 1.28.
2. Since we can't find areas between two values in the standard normal table, we will use the information we know about the values that are to the left of 1.28:
Example 4: Finding the percentage of values between two Z-scores
What percentage of values will lie between -1.28 and 1.28?
1. Draw a diagram: in this example, you are looking for the percentage between a negative and a positive score.
2. The curve is symmetrical. This means that the area between 0 and 1.28 is the same as the area between 0 and -1.28.
3. The percentage of values between 0 and 1.28 is 39.97% (found in example 3). Thus, we will have to multiply 39.97% x 2 = 79.98%
Example 5: Finding Z-scores and raw scores from percentages using the normal curve table.
The table can also be used to find Z-scores and raw scores from specific percentages.
For example, to find the Z-score from the percentage 90%, we look for the most approximate percentage in the table: .8997. Working backwards we see that this figure corresponds to a Z-score of 1.28.
This Z-score can then be converted to a raw score using the mean and the standard deviation of the distribution.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 125, what percentage of people have higher IQs?In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 125, what percentage of people have higher IQs?STEP 1: Convert raw score into Z score.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 125, what percentage of people have higher IQs?STEP 1: Convert raw score into Z score.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 125, what percentage of people have higher IQs?STEP 2: Draw a normal curve: indicate where Z score (1.56) falls and shade the area for which you are finding the percentage.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 125, what percentage of people have higher IQs?STEP 3: Use the table to find percentage. In this case, we are looking for a percentage to the right of a Z score. If we take a look at the table, it will give us the percentage to the left of the Z score 1.56: 0.9406; that is, 94.06%.
z | .00 | .01 | .02 | .03 | .04 | .05 | .06 |
0.0 | .5000 | .5040 | .5080 | .5120 | .5160 | .5199 | .5239 |
0.1 | .5398 | .5438 | .5478 | .5517 | .5557 | .5596 | .5636 |
0.2 | .5793 | .5832 | .5871 | .5910 | .5948 | .5987 | .6026 |
0.3 | .6179 | .6217 | .6255 | .6293 | .6331 | .6368 | .6406 |
0.4 | .6554 | .6591 | .6628 | .6664 | .6700 | .6736 | .6772 |
0.5 | .6915 | .6950 | .6985 | .7019 | .7054 | .7088 | .7123 |
0.6 | .7257 | .7291 | .7324 | .7357 | .7389 | .7422 | .7454 |
0.7 | .7580 | .7611 | .7642 | .7673 | .7704 | .7734 | .7764 |
0.8 | .7881 | .7910 | .7939 | .7967 | .7995 | .8023 | .8051 |
0.9 | .8159 | .8186 | .8212 | .8238 | .8264 | .8289 | .8315 |
1.0 | .8413 | .8438 | .8461 | .8485 | .8508 | .8531 | .8554 |
1.1 | .8643 | .8665 | .8686 | .8708 | .8729 | .8749 | .8770 |
1.2 | .8849 | .8869 | .8888 | .8907 | .8925 | .8944 | .8962 |
1.3 | .9032 | .9049 | .9066 | .9082 | .9099 | .9115 | .9131 |
1.4 | .9192 | .9207 | .9222 | .9236 | .9251 | .9265 | .9279 |
1.5 | .9332 | .9345 | .9357 | .9370 | .9382 | .9394 | .9406 |
If we subtract 94.06% from 100% (total) we obtain 5.94%. Thus, only 5.94% of people have IQ scores higher than 125.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 95, what percentage of people have higher IQs than this person?In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 95, what percentage of people have higher IQs than this person?STEP 1: Convert raw score into Z score.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 95, what percentage of people have higher IQs than this person?STEP 1: Convert raw score into Z score.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 95, what percentage of people have higher IQs than this person?STEP 2: Draw a normal curve: indicate where Z score (-.31) falls and shade the area for which you are finding the percentage.
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
If a person has an IQ of 125, what percentage of people have higher IQs?STEP 3: Use the table to find percentage. In this case, we are looking for a percentage to the right of a Z value. If we take a look at the table, it will give us the percentage to the left of the Z value -.31; that is, .3783
z | .00 | .01 |
-3.4 | .0003 | .0003 |
-3.3 | .0005 | .0005 |
-3.2 | .0007 | .0007 |
-3.1 | .0010 | .0009 |
-3.0 | .0013 | .0013 |
-2.9 | .0019 | .0018 |
-2.8 | .0026 | .0025 |
-2.7 | .0035 | .0034 |
-2.6 | .0047 | .0045 |
-2.5 | .0062 | .0060 |
-2.4 | .0082 | .0080 |
-2.3 | .0107 | .0104 |
-2.2 | .0139 | .0136 |
-2.1 | .0179 | .0174 |
-2.0 | .0228 | .0222 |
-1.9 | .0287 | .0281 |
-1.8 | .0359 | .0351 |
-1.7 | .0446 | .0436 |
-1.6 | .0548 | .0537 |
-1.5 | .0668 | .0655 |
-1.4 | .0808 | .0793 |
-1.3 | .0968 | .0951 |
-1.2 | .1151 | .1131 |
-1.1 | .1357 | .1335 |
-1.0 | .1587 | .1562 |
-0.9 | .1841 | .1814 |
-0.8 | .2119 | .2090 |
-0.7 | .2420 | .2389 |
-0.6 | .2743 | .2709 |
-0.5 | .3085 | .3050 |
-0.4 | .3446 | .3409 |
-0.3 | .3821 | .3783 |
If we subtract 37.83% from 100% (total) we obtain 62.17%. Thus, 62.17% of people have higher IQs than someone with an IQ of 95
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
What IQ score would a person need to be in the top 5%?In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
What IQ score would a person need to be in the top 5%?We will look in the table for the percentage 95% (.9495 in the table), which corresponds to a Z score of 1.64. 95% of scores are to the left of the Z score 1.64.
z | .00 | .01 | .02 | .03 | .04 |
0.0 | .5000 | .5040 | .5080 | .5120 | .5160 |
0.1 | .5398 | .5438 | .5478 | .5517 | .5557 |
0.2 | .5793 | .5832 | .5871 | .5910 | .5948 |
0.3 | .6179 | .6217 | .6255 | .6293 | .6331 |
0.4 | .6554 | .6591 | .6628 | .6664 | .6700 |
0.5 | .6915 | .6950 | .6985 | .7019 | .7054 |
0.6 | .7257 | .7291 | .7324 | .7357 | .7389 |
0.7 | .7580 | .7611 | .7642 | .7673 | .7704 |
0.8 | .7881 | .7910 | .7939 | .7967 | .7995 |
0.9 | .8159 | .8186 | .8212 | .8238 | .8264 |
1.0 | .8413 | .8438 | .8461 | .8485 | .8508 |
1.1 | .8643 | .8665 | .8686 | .8708 | .8729 |
1.2 | .8849 | .8869 | .8888 | .8907 | .8925 |
1.3 | .9032 | .9049 | .9066 | .9082 | .9099 |
1.4 | .9192 | .9207 | .9222 | .9236 | .9251 |
1.5 | .9332 | .9345 | .9357 | .9370 | .9382 |
1.6 | .9452 | .9463 | .9474 | .9484 | .9495 |
In these activities, we will use the distribution of IQ (intelligence quotient) where M= 100 and SD= 16.
What IQ score would a person need to be in the top 5%?Finally, we'll convert the Z score of 1.64 to raw score:
Thus, a person needs an IQ of at least 126.24 to be in the top 5%.