Unit 3 assignment psych stats

Lecture: Z Scores and Probability PSY3200 Unit 3

Defining Z scores

In this unit we will take the information discussed in the previous unit about distributions and variations in scores one step further. There was an example in the previous lecture in which I described if you had taken a biology exam and a psychology exam and in which course were you actually doing better. We determined that in the biology class you were 2 standard deviations above the mean, so you were doing better than you were psychology class because you were only 1 standard deviation above the mean. What we were doing was taking your scores and seeing how they compared to the rest of the distribution. What we were calculating was a standardized score called Z scores (Aron, Coups, & Aron, 2013) .

Z scores are the number of standard deviations you are above or below the mean ( Aron et al., 2013) . In our example from last time we were 1 standard deviation above the average in psychology, so that was a Z score of +1; and we were 2 standard deviations above the mean in biology, so we had a z score of +2. How we calculate these numbers is a simple formula:

M stands for the mean, SD stands for the standard deviation, and X stands for the raw score. To remind you, a raw score is an untreated score, or a score out of context ( Aron et al., 2013) . For example, if I told you that you got a 24 on a quiz you may be disappointed by such a low score; but if the quiz was out of 25, then you would probably be quite happy. The 24 was your raw score, which ends up being a 96%.

We call Z scores standardized scores because as long as we know the mean and standard deviation we can put anything onto the same (standardized) scale ( Aron et al., 2013) . This is the basis of standardized testing that you've probably taken many times throughout your education. Not every student takes the same version of the exam, but as long as educations know the mean and standard deviation of the exams, they can determine where you score compared to every other student.

Z Scores and the Normal Curve

The normal distribution is a well researched curve of data. If we look at the curve (supplied in your unit PowerPoint and text), we see a set of percentages between Z scores ( Aron et al., 2013) . We know that there are approximately 34% of scores that land between the mean (a Z score of 0) and a Z score of +1, also approximately 34% of the population lands between the mean and a Z score of -1 ( Aron et al., 2013). Combined 68% of the population will land between one standard deviation above and below the mean. This makes sense as the peak of the normal curve is towards the middle and the normal curve represents the frequency of the distribution. The higher the part of the curve, the more scores in that region, so it makes sense that more than 2/3 of our scores land in this highest peak of the curve. As you taper out from the middle, approximately 14% lands between Z scores of 1 and 2, and approximately 2% lands between the Z scores of 2 and 3. Again the percentages are symmetrical on each side of the curve, so the same percentages apply to both the positive and negative Z scores ( Aron et al., 2013) . Understanding these approximate percentages will be very helpful in the following topic, however these are just approximate percentages ( Aron et al., 2013) .

In order to find out the exact percentages you need to use the normal curve table that was supplied as an additional resource (Nolan & Heinzen, 2017). Note: there is also a table provided with your textbook ( Aron et al.,

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2013) , however the set of the supplemental table is preferable. As I mentioned the normal curve is a well researched curve of data and this table allows us to calculate the exact percentages between Z scores by defining percentages between Z scores and the mean, and Z scores and the tail of the curve. f you look at the table, it has 3 columns that repeat multiple times. The first column are your Z scores precise to 0.01 for each score; the second column is the percentage of scores between any Z score and the mean (center of the curve); the third column is the percentage of scores between any Z score and the tail (outsides of the curve) (Nolan & Heinzen, 2017). Let's take a Z score of 0 for example. This score is the exact middle of the curve with 50% above and 50% below it. If we check the table, it will say in column 2 that 0% of scores are between that Z score and mean (which makes sense since a Z score of zero is the mean) and third column says 50% which is all the scores between center of the curve and the tail. Let's take one more example before we learn how to solve these problems: if you have a Z score of +1, the percentage to the mean is 34.13%, which makes sense since we knew that between the mean and Z score of 1 was approximately 34%; and the percentage in the tail is 15.87% (which makes sense since the percentage between the Z scores of 1 and 2 is approximately 14% and 2 and 3 is approximately 2% for a combined 16%). The final thing to note on this table is that there are no negative numbers, however this is not a problem since we know the normal curve is symmetrical, negative Z scores will have the same percentages (you just need to be careful which side of the curve you are on for any particular problem).

Calculating Percentages from Raw Scores

Using our knowledge of Z scores and distribution, we can determine the percentage of individuals who scored above or below any particular score. This can be done in 3 steps: (1) convert your raw score to a Z score; (2) draw the normal curve and shade in the area you're looking for; (3) use the provided normal curve table to see the exact percentage you're looking for ( Aron et al., 2013) . The purpose of the second step is that many students are often confused on which column of the table they should use; by drawing it out, it will help to visualize what it is you're looking for.

Let's take a practical example of IQ. IQ has a normal distribution with a M = 100 and a SD = 16. Let's say a person has an IQ of 122, what percentage of scores is above this person? First, we will convert that score to a Z score: X = 122, M = 100, SD = 16. Z = (122-100)/16 = 1.38. Second, we will draw the normal curve and shade in what we are looking for. Draw the curve, indicate the Z score of zero in the middle and our Z score of 1.38 will be to the right of that, somewhere in the middle of the right side. The question asked what is the percentage of scores ABOVE, so everything to the right is above we will shade in the curve to the right of our Z score. The final step is to look up the exact percentage in the table. The area of the curve we just shaded was the tail, so when we look at the table, we fill find our Z score (1.38) and go across to % in tail which gives us the percentage of 8.38%.

Calculating Raw Scores from Percentages

The other thing we can do is find out the exact raw score from a percentage. We can this again in 3 steps: (1) draw the curve and shade in the percentage using the approximate 1-2-3 rule; (2) find the exact percentage and the corresponding Z score on the normal curve table; (3) convert the Z score back to a raw score. In order to do this last step, we need to reverse the Z score formula which will look like this ( Aron et al., 2013) :

For these problems, one thing to be careful of is the side of the curve your Z score is located. The table will give you positive Z scores only so it is your responsibility to realize when you should be using negative Z scores. Let's look at another example of IQ (using the same mean and standard deviation): What IQ would you need to be in

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the lowest 15%? Step 1 is to draw the curve and shade; since we are asking for the lowest 15% we would shade on the left side of the curve from the middle of the left side towards to the end of the tail, so we have shaded in the percentage to the tail. Step 2 is to look up the exact percentage in the table. You won't often find the exact percentage so as a general rule we will find the percentage closest to it. In this case the percentage closest to 15% is 14.92%. The Z score that goes with that is 1.04. We would then convert this back to a raw score however before we do that, we must recognize that we are on the left side of the curve which is the negative side of the curve. So, our Z score is -1.04. Finally we will convert back to a raw score: Z = -1.04, M = 100, SD = 16; X = (-1.04*16)+100 = 83.36.

A few notes to be careful of when doing these problems: (1) sometimes you will be asked to solve for percentages larger than the table can give you, in these cases you will need to solve the problem in pieces for example if I am asked to solve for the percentage of scores above a raw score that is less than the mean, that will be more than 50% of the distribution (and the table only gives you up to 50%). In this case you'd solve for the percentage to the mean and then add the remaining 50%. Next (2) you may be asked to solve for a range of raw scores. In this case you would follow the same steps as converting from a percentage to a raw score, but you'd have to do it twice (once for each Z score) to get the full range. Finally, (3) you may be asked to solve for a percentile, which is different from a percentage. A percentile is the full distribution ranging from the most left of the curve (0th percentile) to the far right (100th percentile). If I ask you for the raw score at the 80th percentile, I would be asking for 30% above the average so you would solve looking for the raw score either 30% to the mean or 20% to the tail.

Probability and Populations

Finally, the last thing to discuss in this unit is the probability. Probability is the likelihood an event will occur ( Aron et al., 2013) . For example, if I ask for the probability of getting a 2 on a roll of a die, you would take the possible correct outcomes and divide by the total possible outcomes. In this case there is 1 correct outcome of getting a 2, while there are 6 possible outcomes of the numbers 1-6. You would divide 1/6 = 0.17 or 17%. The rules of probability are outlined in our text as well as techniques to add or multiply different probability types ( Aron et al., 2013) . Probability will also play a large role in our next topic of hypothesis testing. The text continues describing populations and sampling (something covered in our previous unit), as well as defining symbols between samples and populations ( Aron et al., 2013) .

(CSLO 1, CSLO 2, CSLO 3, CSLO 6, CSLO 11)

References

Aron, A. Coups, E.J. & Aron, E. (2013) Statistics for Psychology (6th ed.) Chapter 3. Nolan, S. & Heinzen, T. (2017) Statistics for the Behavioral Sciences (4th ed.) Appendix B.

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