Z-Scores and the Normal Curve PDF

Z Scores and the Normal Curve  What is a Z Score? – A Z score makes use of the mean and standard deviation to describe a particular score. Specifically, a Z score is the number of standard deviations the actual score is above or below the mean. If the actual score is above the me...

Z Scores and the Normal Curve  What is a Z Score? – A Z score makes use of the mean and standard deviation to describe a particular score. Specifically, a Z score is the number of standard deviations the actual score is above or below the mean. If the actual score is above the mean, the Z score is positive. If the actual score is below the mean, the Z score is negative. The standard deviation now becomes a kind of yardstick, a unit of measure in its own right.  For example: – Jerome has a score of 5, which is 1.60 units above the mean of 3.40. One standard deviation is 1.47 units; so Jerome’s score is a little more than 1 standard deviation above the mean. To be precise, Jerome’s Z score is +1.6 (that is, his score of 5 is 1.6 standard deviations above the mean).  Z Scores as a Scale – 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. Changing a number to a Z score is a bit like converting words for measurement in various obscure languages into one language that everyone can understand— inches, cubits; for example, into centimeters. It is a very valuable tool.  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). – Suppose Ryan were observed speaking to other children 20 times in an hour. Ryan would clearly be unusually talkative, with a Z score of +2. Ryan speaks not merely more than the average but more by twice as much as children tend to vary from the average!  Formula to Change a Raw Score to a Z Score – A Z score is the number of standard deviations by which the raw score is above or below the mean. To figure a Z score, subtract the mean from the raw score, giving the deviation score. Then divide the deviation score by the standard deviation. The formula is: – For example, using the formula for Jacob, the child who spoke to other children 8 times in an hour (where the mean number of times children speak is 12 and the standard deviation is 4),  Formula to Change a Z Score to a Raw Score – To change a Z score to a raw score, the process is reversed: multiply the Z score by the standard deviation and then add the mean. The formula is: – 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,  The Mean and Standard Deviation of Z Scores – The mean of any distribution of Z scores is always 0. This is so because when you change each raw score to a Z score, you take the raw score minus the mean. So the mean is subtracted out of all the raw scores, making the overall mean come out to 0. In other words, in any distribution, the sum of the positive Z scores must always equal the sum of the negative Z scores. Thus, when you add them all up, you get 0.  The Mean and Standard Deviation of Z Scores – The standard deviation of any distribution of Z scores is always 1. This is because when you change each raw score to a Z score, you divide the score by one standard deviation. As you work through the examples that follow, this will become increasingly intuitive. It is important to note that the shape of a distribution is not changed when raw scores are converted to Z scores. So, for example, if a distribution of raw scores is positively skewed, the distribution of Z scores will also be positively skewed.  Note: – A Z score is sometimes called a standard score. There are two reasons: Z scores have standard values for the mean and the standard deviation, and, as we saw earlier, Z scores provide a kind of standard scale of measurement for any variable. (However, sometimes the term standard score is used only when the Z scores are for a distribution that follows a normal curve.)  The Normal Curve – The normal curve is a mathematical (or theoretical) distribution. Researchers often compare the actual distributions of the variables they are studying (that is, the distributions they find in research studies) to the normal curve. – They don’t expect the distributions of their variables to match the normal curve perfectly (since the normal curve is a theoretical distribution), but researchers often check whether their variables approximately follow a normal curve. – The normal curve or normal distribution is also often called a Gaussian distribution after the astronomer Karl Friedrich Gauss. However, if its discovery can be attributed to anyone, it should really be to Abraham de Moivre.  A normal curve.  The Normal Curve and the Percentage of Scores Between the Mean and 1 and 2 Standard Deviations from the Mean – The shape of the normal curve is standard. 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, approximately 34% of the scores are always between the mean and 1 standard deviation from the mean. – Consider IQ scores. On many widely used intelligence tests, the mean IQ is 100, the standard deviation is 15, and the distribution of IQs is roughly a normal curve. – Knowing about the normal curve and the percentage of scores between the mean and 1 standard deviation above the mean tells you that about 34% of people have IQs between 100, the mean IQ, and 115, the IQ score that is 1 standard deviation above 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% (34% + 34%) have IQs between 85 and 115.  Distribution of IQ scores on many standard intelligence tests (with a mean of 100 and a standard deviation of 15). – 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. (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). – 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.  Example: – 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.  The Normal Curve Table and Z Scores – The 50%, 34%, and 14% figures are important practical rules for working with a group of scores that follow a normal distribution. – However, in many research and applied situations, psychologists need more accurate information. Because the normal curve is a precise mathematical curve, you can figure the exact percentage of scores between any two points on the normal curve (not just those that happen to be right at 1 or 2 standard deviations from the mean).  The Normal Curve Table and Z Scores – For example, exactly 68.59% of scores have a Z score between +.62 and -1.68; exactly 2.81% of scores have a Z score between +.79 and +.89; and so forth. Statisticians have worked out tables for the normal curve that give the percentage of scores between the mean (a Z score of 0) and any other Z score (as well as the percentage of scores in the tail for any Z score).  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? Example 2: If a person has an IQ of 95, what percentage of people have higher IQs? IQ=125 M=100 SD=15 𝑥−𝑀 125−100 Z= = = 1.67 𝑆𝐷 15  𝑥−𝑀 125−100 Z= = = 1.67 ≈ 0.9525 𝑆𝐷 15 Or 95.25% 1.67 34% ??? 2.5% 13.5% 95.25% If a person has an IQ of 125, what percentage of people have higher IQs? 1.67 34% 4.75 2.5% 13.5% 95.25%  Examples: – Here are two examples using IQ scores where M = 100 and SD = 15. Example 1: What IQ score would a person need to be in the top 5%? Example 2: Example 2: What IQ score would a person need to be in the top 55%? Example 3: What range of IQ scores includes the 95% of people in the middle range of IQ scores?  Exercise 1 – An entrance examination to be conducted to 120 incoming freshmen students at a certain university which is known to be normally distributed and has a mean of 85 and standard deviation of 10 a) what proportion of the students would be expected to score above 95? b) What percent of students would be expected to score above 77? c) What is the probability of students that would be expected to score below 110? d) How many students would be expected to score below 90? e) How many of the students would be expected to score between 80 and 98? f) What should be the maximum possible score for a student to be bellow the 31.56%? Exercise 2 1. Why is the normal curve (or at least a curve that is symmetrical and unimodal) so common in nature? 2. 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? 3. 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? 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%? 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? 6. Using the normal curve table, what Z score would you have if (a) 20% are above you and (b) 80% are below you?