Chapter 3 Statistics Lecture
A Z score is sometimes called a
standard score. For two reasons: Z scores have standard values for the mean and the SD, and, as we saw earlier, Z scores provide a kind of standard scale of measurement for any variable. However, sometimes the term standard curve is used only when the Z scores are for a distribution that follows a normal curve.
Normal distribution is
the bell-shaped smooth histograms aka the frequency distribution that follows a normal curve
A z score is defined as
the number of standard deviations that a score is above (or below, if it is negative) the mean of its distribution; it is thus an ordinary score transformed so that it better describes the score's location in a distribution.
It's important to note that
the shape of a distribution is not changed when raw scores are converted to Z scores.
1. Why is the normal curve (or at least a curve that is symmetrical and unimodal) so common in nature
. It is common because any particular score is the result of the random combination of many effects, some of which make the score larger and some of which make the score smaller. Thus, on average these effects balance out near the middle, with relatively few at each extreme, because it is unlikely for most of the increasing and decreasing effects to come out in the same direction
Example of Normal curve with approximate percentages of scores between the mean and 1 and 2 standard deviations above and below the mean.
Similarly, because the normal curve is symmetrical, about 34% of people have IQs between 100 and 85 (the score that is 1 standard deviation below the mean), and 68% 134% + 34%2 have IQs between 85 and 115. There are many fewer scores between 1 and 2 standard deviations from the mean than there are between the mean and 1 standard deviation from the mean. It turns out that about 14% of the scores in a normal curve are between 1 and 2 standard deviations above the mean (see Figure 3-7). (Similarly, about 14% of the scores are between 1 and 2 standard deviations below the mean.) Thus, about 14% of people have IQs between 115 (1 standard deviation above the mean) and 130 (2 standard deviations above the mean)
The standard deviation of any distribution of Z scores is
always 1
Remember that negative Z scores are scores below the mean,
and positive Z scores are scores above the mean.
the mean of any distribution of Z scores is always
0.
Without using a normal curve table, about what percentage of scores on a normal curve are (a) between the mean and 2 SDs above the mean, (b) below 1 SD above the mean, (c) above 2 SDs below the mean?
3. (a) Between the mean and 2 SDs above the mean: 48%; (b) below 1 SD above the mean: 84%; (c) above 2 SDs below the mean: 98%.
4. Without using a normal curve table, about what Z score would a person have who is at the start of the top (a) 50%, (b) 16%, (c) 84%, (d) 2%?
4. (a) 50%: 0; (b) 16%: 1; (c) 84%: -1; (d) 2%: 2.
5. Using the normal curve table, what percentage of scores are (a) between the mean and a Z score of 2.14, (b) above 2.14, (c) below 2.14?
5. (a) Between the mean and a Z score of 2.14: 48.38%; (b) above 2.14: 1.62%; (c) below 2.14: 98.38%.
Thus, there is a known percentage of scores above or below any particular point.
For example, exactly 50% of the scores in a normal curve are below the mean, because in any symmetrical distribution half the scores are below the mean. More interestingly, as shown in Figure 3-7, approximately 34% of the scores are always between the mean and 1 standard deviation from the mean.
Example of z score
Jerome has a score of 5 on the scale from 1.7, the standard deviation is 1.47 and the meaning rating is 3.40; however, since we have to do (X-M) we do 5-3.40 which leads us to 1.60, which is a bit more than the average students typically get. To be precise, Jerome's Z score is +1.09 (that is, his score of 5 is 1.09 standard deviations above the mean). Another student, Ashley, has a score of 2. Her score is 1.40 units below the mean. Therefore, her score is a little less than 1 standard deviation below the mean (a Z score of -.95). So, Ashley's score is below the average by about as much as students typically vary from the average
example:
Here are three examples. Once again, we use IQ for our examples, with M = 100 and SD = 15. Example 1: What IQ score would a person need to be in the top 5%? ❶ Draw a picture of the normal curve, and shade in the approximate area for your percentage using the 50%-34%-14% percentages. We wanted the top 5%. Thus, the shading has to begin above (to the right of) 1 SD (there are 16% of scores above 1 SD). However, it cannot start above 2 SD because only 2% of all the scores are above 2 SD. But 5% is a lot closer to 2% than to 16%. Thus, you would start shading a small way to the left of the 2 SD point. This is shown in Figure 3-12. Make a rough estimate of the Z score where the shaded area stops. The Z score is between +1 and +2. ❸ Find the exact Z score using the normal curve table (subtracting 50% from your percentage if necessary before looking up the Z score). We want the top 5%, which means we can use the "% in Tail" column of the normal curve table. Looking in that column, the closest percentage to 5% is 5.05% (or you could use 4.95%). This goes with a Z score of 1.64 in the "Z" column. ❹ Check that your exact Z score is within the range of your rough estimate from Step ❷. As we estimated, +1.64 is between +1 and +2 (and closer to 2). ❺ If you want to find a raw score, change it from the Z score. Using the formula, X = 1Z21SD2+ M = 11.6421152+ 100 = 124.60. In sum, to be in the top 5%, a person would need an IQ of at least 124.60 Example 2: What IQ score would a person need to be in the top 55%? Draw a picture of the normal curve and shade in the approximate area for your percentage using the 50%-34%-14% percentages. You want the top 55%. There are 50% of scores above the mean. So, the shading has to begin below (to the left of) the mean. There are 34% of scores between the mean and 1 SD below the mean; so the score is between the mean and 1 SD below the mean. You would shade the area to the right of that point. This is shown in Figure 3-13. Figure 3-13 Finding the IQ score for where the top 55% of scores start. ❷ Make a rough estimate of the Z score where the shaded area stops. The Z score has to be between 0 and -1. ❸ Find the exact Z score using the normal curve table (subtracting 50% from your percentage if necessary before looking up the Z score). Being in the top 55% means that 5% of people have IQs between this IQ and the mean (that is, 55% - 50% = 5%). In the normal curve table, the closest percentage to 5% in the "% Mean to Z" column is 5.17%, which goes with a Z score of .13. Because you are below the mean, this becomes -.13. ❹ Check that your exact Z score is within the range of your rough estimate from Step ❷. As we estimated, -.13 is between 0 and -1. ❺ If you want to find a raw score, change it from the Z score. Using the usual formula, X = 1-.1321152 + 100 = 98.05. So, to be in the top 55% on IQ, a person needs an IQ score of 98.05 or higher. Example 3: What range of IQ scores includes the 95% of people in the middle range of IQ scores? This kind of problem, of finding the middle percentage, may seem odd at first. However, it is actually a very common situation used in procedures you will learn in later chapters. Think of this kind of problem in terms of finding the scores that go with the upper and lower ends of this percentage. Thus, in this example, you are trying to find the points where the bottom 2.5% ends and the top 2.5% begins (which, out of 100%, leaves the middle 95%). ❶ Draw a picture of the normal curve, and shade in the approximate area for your percentage using the 50%-34%-14% percentages. Let's start where the top 2.5% begins. This point has to be higher than 1 SD (16% of scores are higher than 1 SD). However, it cannot start above 2 SD because there are only 2% of scores above 2 SD. But 2.5% is very close to 2%. Thus, the top 2.5% starts just to the left of the 2 SD point. Similarly, the point where the bottom 2.5% comes in is just to the right of -2 SD. The result of all this is that we will shade in two tail areas on the curve: one starting just above -2 SD and the other starting just below +2 SD. This is shown in Figure 3-14. ❷ Make a rough estimate of the Z score where the shaded area stops. You can see from the picture that the Z score for where the shaded area stops above the mean is just below +2. Similarly, the Z score for where the shaded area stops below the mean is just above -2. ❸ Find the exact Z score using the normal curve table (subtracting 50% from your percentage if necessary before looking up the Z score). Being in the top 2.5% means that 2.5% of the IQ scores are in the upper tail. In the normal curve table, the closest percentage to 2.5% in the "% in Tail" column is exactly Figure 3-14 Finding the IQ scores for where the middle 95% of scores begins and ends.2.50%, which goes with a Z score of +1.96. The normal curve is symmetrical. Thus, the Z score for the lower tail is -1.96. ❹ Check that your exact Z score is within the range of your rough estimate from Step ❷. As we estimated, +1.96 is between +1 and +2 and is very close to +2, and -1.96 is between -1 and -2 and very close to -2. ❺ If you want to find a raw score, change it from the Z score. For the high end, using the usual formula, X = 11.9621152 + 100 = 129.40. For the low end, X = 1-1.9621152 + 100 = 70.60. In sum, the middle 95% of IQ scores run from 70.60 to 129.40
You will find it very useful to remember the 34% and 14% figures. These figures tell you the percentages of people above and below any particular score whenever you know that score's number of standard deviations above or below the mean
You can also reverse this approach and figure out a person's number of standard deviations from the mean from a percentage. Suppose you are told that a person scored in the top 2% on a test. Assuming that scores on the test are approximately normally distributed, the person must have a score that is at least 2 standard deviations above the mean. This is because a total of 50% of the scores are above the mean, but 34% are between the mean and 1 standard deviation above the mean, and another 14% are between 1 and 2 standard deviations above the mean. That leaves 2% of scores (that is, 50% - 34% - 14% = 2%) that are 2 standard deviations or more above the mean.
Normal curve table
e table showing percentages of scores associated with the normal curve; the table usually includes percentages of scores between the mean and various numbers of standard deviations above the mean and percentages of scores more positive than various numbers of standard deviations above the mean
So, for example,
if a distribution of raw scores is positively skewed, the distribution of Z scores will also be positively skewed.
a Z score describes a score
in terms of how much it is above or below the average.
A raw score
is an ordinary score as opposed to a Z score. The two scales are something like a ruler with inches lined up on one side and centimeters on the other
Normal curve
is specific, mathematically defined, bell-shaped frequency distribution that is symmetrical and unimodal; distributions observed in nature and in research commonly approximate it
The shape of the normal curve
is standard
Steps to change a z score to a raw score (example) : ❶ Figure the deviation score: multiply the Z score by the standard deviation. ❷ Figure the raw score: add the mean to the deviation score. Suppose a child has a Z score of 1.5 on the number of times spoken with another child during an hour. This child is 1.5 standard deviations above the mean. Because the standard deviation in this example is 4 raw score units (times spoken), the child is 6 raw score units above the mean, which is 12. Thus, 6 units above the mean is 18. Using the formula
1.5(4)+12=6+12=18
formula for z score is
X-M/ SD 8-8=0/0
Formula to Change a Z score to a Raw score
X= (Z)(SD)+M
. For a particular group of scores, M = 10 and SD = 2. Give the raw score for a Z score of (a) +2, (b) +.5, (c) 0, and (d) -3.
Z*SD+M 2*2+10=14 .5*2=1+10=11 0*2+10=10 -3*2=-6+10=4
Write the formula for changing a Z score to a raw score, and define each of the symbols.
Z*SD+M Z= Z score SD= standard deviation M=Mean
For a particular group of scores, M = 20 and SD = 5. Give the Z score for (a) 30, (b) 15, (c) 20, and (d) 22.5
Z= X-M/SD 30-20/5=2 15-20=-5/5=-1 20-20=0/5=0 22.5-20=2.5/5=0.5
Write the formula for changing a raw score to a Z score, and define each of the symbols.
Z= X-M/SD Z= Z score X= score SD= Standard deviation
Figuring Z Scores and Raw Scores from Percentages Using the Normal Curve Table Going from a percentage to a Z score or raw score is similar to going from a Z score or raw score to a percentage. However, you reverse the procedure when figuring the exact percentage. Also, any necessary changes from a Z score to a raw score are done at the end. Here are the five steps
❶ Draw a picture of the normal curve, and shade in the approximate area for your percentage using the 50%-34%-14% percentages. ❷ Make a rough estimate of the Z score where the shaded area stops. ❸ Find the exact Z score using the normal curve table (subtracting 50% from your percentage if necessary before looking up the Z score). Looking at your picture, figure out either the percentage in the shaded tail or the percentage between the mean and where the shading stops. For example, if your percentage is the bottom 35%, then the percentage in the shaded tail is 35%. Figuring the percentage between the mean and where the shading stops will sometimes involve subtracting 50% from the percentage in the problem. For example, if your percentage is the top 72%, then the percentage from the mean to where that shading stops is 22% 172% - 50% = 22%2. Once you have the percentage, look up the closest percentage in the appropriate column of the normal curve table ("% Mean to Z" or "% in Tail") and find the Z score for that percentage. That Z will be your answer—except it may be negative. The best way to tell if it is positive or negative is by looking at your picture. ❹ Check that your exact Z score is within the range of your rough estimate from Step ❷. ❺ If you want to find a raw score, change it from the Z score. Use the usual formula, X = (Z)(SD)+M
Steps to change a raw score to a z score example: Suppose that a developmental psychologist observed 3-year-old Jacob in a laboratory situation playing with other children of the same age. During the observation, the psychologist counted the number of times Jacob spoke to the other children. The result, over several observations, is that Jacob spoke to other children about 8 times per hour of play. Without any standard of comparison, it would be hard to draw any conclusions from this. Let's assume, however, that it was known from previous research that under similar conditions, the mean number of times children speak is 12, with a standard deviation of 4. With that information, we can see that Jacob spoke less often than other children in general, but not extremely less often. Jacob would have a Z score of -1 (M = 12 and SD = 4, thus a score of 8 is 1 SD below M), as shown in Figure 3-3.
1. FIGURE THE DEVIATION SCORE: subtract the mean from the raw score. 8-12=-4 2. FIGURE THE Z SCORE: DIVIDE THE DEVIATION SCORE BY THE SD -4/4=-1
Without using a normal curve table, about what percentage of scores on a normal curve are (a) above the mean, (b) between the mean and 1 SD above the mean, (c) between 1 and 2 SDs above the mean, (d) below the mean, (e) between the mean and 1 SD below the mean, and (f) between 1 and 2 SDs below the mean?
2. (a) Above the mean: 50%; (b) between the mean and 1 SD above the mean: 34%; (c) between 1 and 2 SDs above the mean: 14%; (d) below the mean: 50%; (e) between the mean and 1 SD below the mean: 34%; (f) between 1 and 2 SDs below the mean: 14%
Using the normal curve table, what Z score would you have if (a) 20% are above you and (b) 80% are below you?
6. (a) 20% above you: .84; (b) 80% below you: .84.
How is a Z score related to a raw score.
To find a raw score you have to do Z*SD+ M and a Z score is the number of standard deviations a raw score is above or below the mean.
Suppose a person has a Z score for overall health of +2 and a Z score for overall sense of humor of +1. What does it mean to say that this person is healthier than she is funny?
Ultimately, this shows that the person is above average in health than she is above the average in humor.
This creates a unimodal distribution with most of the scores near the middle and fewer at the extremes. It also creates a distribution that is symmetrical, because the number of letters recalled is as likely to be above as below the middle. Being a unimodal symmetrical curve does not guarantee that it will be a normal curve; it could be too flat or too pointed. However, it can be shown mathematically that in the long run, if the influences are truly random, and the number of different influences being
combined is large, a precise normal curve will result. Mathematical statisticians call this principle the central limit theorem. We have more to say about this principle in Chapter 5.
Steps for Figuring the Percentage of Scores Above or Below a Particular Raw Score or Z Score Using the Normal Curve Table
❶ If you are beginning with a raw score, first change it to a Z score. Use the usual formula, Z = (X - M)/SD ❷ Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage. When marking where the Z score falls on the normal curve, be sure to put it in the right place above or below the mean according to whether it is a positive or negative Z score ❸ Make a rough estimate of the shaded area's percentage based on the 50%- 34%-14% percentages. You don't need to be very exact; it is enough just to estimate a range in which the shaded area has to fall, figuring it is between two particular whole Z scores. This rough estimate step is designed not only to help you avoid errors (by providing a check for your figuring), but also to help you develop an intuitive sense of how the normal curve works ❹ Find the exact percentage using the normal curve table, adding 50% if necessary. Look up the Z score in the "Z" column of Table A-1 and find the percentage in the "% Mean to Z" column or "% in Tail" column next to it. If you want the percentage of scores between the mean and this Z score, or if you want the percentage of scores in the tail for this Z score, the percentage in the table is your final answer. However, sometimes you need to add 50% to the percentage in the table. You need to do this if the Z score is positive and you want the total percentage below this Z score, or if the Z score is negative and you want the total percentage above this Z score. However, you don't need to memorize these rules. It is much easier to make a picture for the problem and reason out whether the percentage you have from the table is correct as is, or if you need to add 50%. ❺ Check that your exact percentage is within the range of your rough estimate from Step ❸.
Examples Here are two examples using IQ scores where M = 100 and SD = 15. Example 1: If a person has an IQ of 125, what percentage of people have higher IQs? Distribution of IQ scores showing percentage of scores above an IQ score of 125 (shaded area).
❶ If you are beginning with a raw score, first change it to a Z score. Using the usual formula, Z = (X - M2)/SD, Z = (125 - 100)/15 = +1.67. ❷ Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage. This is shown in Figure 3-10 (along with the exact percentages figured later). ❸ Make a rough estimate of the shaded area's percentage based on the 50%- 34%-14% percentages. If the shaded area started at a Z score of 1, it would have 16% above it. If it started at a Z score of 2, it would have only 2% above it. So, with a Z score of 1.67, the number of scores above it has to be somewhere between 16% and 2%. ❹ Find the exact percentage using the normal curve table, adding 50% if necessary. In Table A-1, 1.67 in the "Z " column goes with 4.75 in the "% in Tail" column. Thus, 4.75% of people have IQ scores higher than 125. This is the answer to our problem. (There is no need to add 50% to the percentage.) ❺ Check that your exact percentage is within the range of your rough estimate from Step ❸. Our result, 4.75%, is within the 16-to-2% range we estimated. Example 2: If a person has an IQ of 95, what percentage of people have higher IQs? ❶ If you are beginning with a raw score, first change it to a Z score. Using the usual formula, Z = (95 - 100)/15 = -.33. ❷ Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage. This is shown in Figure 3-11 (along with the percentages figured later). Distribution of IQ scores showing percentage of scores above an IQ score of 95 (shaded area). ❸ Make a rough estimate of the shaded area's percentage based on the 50%- 34%-14% percentages. You know that 34% of the scores are between the mean and a Z score of -1. Also, 50% of the curve is above the mean. Thus, the Z score of -.33 has to have between 50% and 84% of scores above it. ❹ Find the exact percentage using the normal curve table, adding 50% if necessary. The table shows that 12.93% of scores are between the mean and a Z score of .33. Thus, the percentage of scores above a Z score of -.33 is the 12.93% between the Z score and the mean plus the 50% above the mean, which is 62.93%. ❺ Check that your exact percentage is within the range of your rough estimate from Step ❸. Our result of 62.93% is within the 50-to-84% range we estimated.
