Chapter 7, "The Normal Probability Distribution,"

If you wanted to find the proportion of area that falls above z = 1.3 you would find the proportion that falls below z = 1.3 and subtract it from 1.

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The z-distribution has a mean of 0 and a standard deviation of 1.

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The area between the mean and Z for a z-score of 1.0 is

0.3413

Standard Normal Distribution

Special case of normal probability distribution with a mean of 0 and a standard deviation of 1.

1. Normal Distribution is a continuous probability distribution for a random variable, x. 2. The graph of the normal distribution is called a Normal Curve . 3. The mean and the Standard Deviation of a distribution determine the shape of the normal curve. 4. The smaller the standard deviation, the less spread out the data are. 5. The larger the standard deviation, the more spread out the data are.

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The z-Table

Once we have z-scores, we can use the standard normal table to find out the area under the normal curve that corresponds to that z-score. To use the table, scroll down the columns with z at the top (there are several - just keep scrolling down until you find the z value you are looking for), then look to the column immediately to the right to see how much area under the curve falls between the mean and that z-score. The column next to that one indicates the proportion of the area that falls beyond that z-score. Because the normal curve is symmetrical, the table works for both positive and negative z-scores, that is, the area under the curve is the same for both the positive and negative forms of a particular z-score. You can calculate probabilities using the z-table and z-scores. Because the normal curve, or bell curve, is symmetrical, the area under the curve is the same for both the positive and negative values of each z-score. You would use the same value for a z-score of -0.79 as you would for 0.79.

Area Above the z-Score: Perhaps we want to know what proportion of the area of the curve falls above z = 1.25.

Step 1: Sketch the curve and shade the area we are trying to find. Step 2: The first column of the table will give us the proportion of the area between the mean and z = 1.25, but that's not what we're interested in. The second column will give us the proportion of the area under the curve that falls beyond z = 1.25. That's what we want! So the proportion of the area under the curve that falls beyond z = 1.25 is .1056.

Area Between Two z-Scores: What if we want to know what proportion of the area falls between z = -1.35 and z = .75? We follow similar steps as our earlier examples.

Step 1: Sketch the curve and shade the area we are trying to find. Step 2: Use the table to find the area under the curve between the mean and z = -1.35 and between the mean and z = .75. z = -1.35: .4115 z = .75: .2734 Step 3: Add together the two proportions to get the total area of the curve that falls between z = -1.35 and z = .75 .4115 + .2734 = .6849

There are several characteristics that hold true for ALL normal curves:

The normal curve is symmetrical, that is, the right half is the mirror image of the left half. Half of the observations fall above the mean and half fall below the mean. Most of the observations cluster around the mean; there are fewer observations as the curve moves away from the mean. The mean, median, and mode are equal, that is, they are the same value. A normal curve approaches but never touches the x-axis. The total area under a normal curve is equal to 1.

There are some other areas that are known to be true for all normal curves:

The proportion of the area under the curve that falls between the mean and 1 standard deviation is .3413. So 68% of our observations will fall between -1 and 1 standard deviations of the mean. The proportion of the area under the curve that falls between the first and the second standard deviations is .1359. Thus, 95.44% of the observations fall within two standard deviations of the mean (-2 to 2 standard deviations). The proportion of the area under the curve that falls between the second and third standard deviations is .0214. Thus, 99.72% of all observations fall within three standard deviations of the mean (-3 to 3 standard deviations). Only .26% of the observations fall between 3 and 4 standard deviations from the mean, with the remaining .02% falling beyond 4 standard deviation units from the mean (-4 and 4).

Calculating Raw Scores

We can also work backwards to calculate a raw score from a z-score. x space equals space X with bar on top space plus space z s Given a student's test grade in the form of a z-score, we can convert it back to the raw score using this formula. Let's revisit Sue, and her z = .29 in Algebra. We know that the class mean was 84 and the standard deviation was 3.45. Plug those values into the formula to determine Sue's raw score. x = 84 + .29(3.45) x = 84 + 1.0005 = 85.0005 = 85 Sue's score in Algebra was an 85.

Normal curves

are used in education, scientific fields, and even for employee evaluations. - All normal curves are "bell-shaped," but some normal curves are thinner and taller while others are shorter and wider. The spread of the data in relation to the mean determines how spread out the curve is.

The standard normal table lists the area under the normal curve that falls

between the mean and a given z-score and beyond a given z-score

The total area under a normal curve is equal to 1. A normal curve approaches, but never touches, the x-axis.

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An infinite number of normal distributions are possible. The standard normal distribution is a distribution of standard scores, also known as z-scores. By calculating z-scores from raw data, you can use those scores to compare data from different situations. The standard normal table, or z-table, displays the area under the curve for any given z-value.

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The normal distribution is a continuous probability distribution for a random variable x. The graph of the normal distribution is a normal curve, or bell curve. Whether the curve is thin and tall or short and wide depends on whether the standard deviation is small or large.

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The standard normal table, or z-table, consists of three columns: Column 1: The z-score Column 2: The area under the curve between the mean and the z-score Column 3: The proportion of the area that falls beyond that z-score To find the proportion of the area that falls between two z-scores, find the proportions for both z-scores and subtract them. You always subtract the smaller number from the larger number. Remember, area will always be positive.

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Normal Distribution

A continuous probability distribution for a random variable, x. - The normal distribution is the continuous probability distribution for a random variable, x. The graph of the normal distribution is called a normal curve, or a bell curve.

Continuous Probability Distribution

A continuous probability distribution is a probability distribution of a continuous random variable. Continuous random variables have an uncountable number of possible outcomes, including fractions and decimals, represented by an interval on the number line. - A probability distribution of a continuous random variable.

Mean and Standard Deviation

The mean and the standard deviation of a distribution determine the shape of the normal curve. While all normal curves will appear "bell-shaped," some normal curves are thinner and taller while others are shorter and wider. The smaller the standard deviation, the less spread out the data are. Thus, the narrower the normal curve will be. Remember that the total area under the curve is equal to 1.

The z-Score

The z-score is a standard score with a mean of 0 and a standard deviation of 1. The use of z-scores allows us to compare data from one situation to data from another situation. The z-score distribution is a standard normal distribution. The z-score calculated for an observation equals the number of standard deviations that the given observation, or x, falls from the mean. We can take data that we collect, convert it to z-scores, and compare those z-scores to answer our questions. The formula for calculating z is: where xis our raw data value or observation, x with bar on top is the mean of the group that the x came from, and s is the standard deviation for that group.

Using the z-Table

We can use the table to determine how much area under the curve is captured by the z-scores that we calculate. Using our z-scores from Sue's classes, we find that the area under the curve between the mean and z = .29 is 0.1141 and that the area under the curve between the mean and z = -.67 is 0.2486.

Bell curve or normal curve

shows the normal distribution of a set of data. Most scores cluster around the mean, and half the data fall above the mean and half below the mean. Only a few scores will be extremely high or extremely low. *The normal distribution is one of the most important distributions in probability theory. The graph of the normal distribution is called a normal curve, or a bell curve.*

Calculating the z-Score

For our example started earlier, where we are trying to determine in which class Sue is better performing, we would first need to calculate Sue's z-score for each class and then compare the z-scores to answer the question. Now, compare the z-scores to determine in which class Sue is performing better. Remember that the z-score equals the number of standard deviations a given value x falls from the mean. Positive z-scores indicate that x falls above the mean; negative z-scores indicate that x falls below the mean. In Algebra, Sue falls above the mean since her z =.29 but in English, Sue falls below the mean since her z = -.67. So, even though Sue's average is higher in English, she is performing better in Algebra, and we know that because her z-score is higher for Algebra than for English. However, it should be evident to you that z-scores are very dependent on the mean and standard deviation of the group from which x comes. *The z-score is a standard score with a mean of 0 and a standard deviation of 1. You collect raw data values and use the formula to convert them to z-scores to compare information.* *where x is the raw data value, X with bar on top is the mean of the group that x came from, and s is the standard deviation for that group.*

Area Below the <span class="italic">z</span>-Score: Perhaps we want to know what proportion of the area falls at or below z = 0.25.

Step 1: Sketch the curve and shade the appropriate area under the curve. Step 2: Use the table to find the area that corresponds to z = 0.25. The proportion of the area under the curve that falls between the mean and z = 0.25 is 0.0987. This is not our answer!! This is only the area between the mean and our z-score. We want to know what proportion of the area under the curve falls at or below z = 0.25. Step 3: Remember what we know about our curve. The normal curve is symmetrical, and half of the area falls above the mean and half falls below. So, in order to find our answer, we must add .50 (which represents the left half of the curve, that is, everything below the mean) to the .0987 that we found in the table (which represents the area from the mean to z = 0.25). Step 4: Our final answer, then, is .5987. The proportion of the area under the curve that falls at or below z = 0.25 is .5987 or .60 (if we round our answer to two decimal places).

Another Example What if we want to know what proportion of the area under the curve falls between z = -2.25 and z = -1.25?

Step 1: Sketch the curve and shade the area we are trying to find. Step 2: Use the table to find the area under the curve between the mean and z = -2.25 and between the mean and z = -1.25. z = -2.25: .4878 z = -1.25: .3944 Step 3: We now have to subtract the smallest area from the largest area to get our answer, because we're not interested in that chunk of area between the mean and z = -1.25. .4878 - .3944 = .0934 Area cannot be a negative number!! You will always subtract the smallest number from the largest number, and your answer will always be positive.

Standard scores

The number of standard deviations a given value x falls from the mean µ. It is also called z-score. -allow comparisons between groups that would not normally be comparable. Scores are standardized so that they are based on the same normal distribution. - There are an infinite number of normal distributions, each with its own mean and standard deviation. The standard normal distribution is a special case, with a set mean and a set standard deviation. The standard normal distribution is a distribution of standard scores. There are several ways to standardize scores. It requires that you convert the raw data to a scale with a set mean and a set standard deviation. For example, most standardized tests are standard scores: there's a set mean and a set standard deviation onto which all raw scores are rescaled. The IQ test, for example, has a mean of 100 and a standard deviation of 15. All raw scores collected from IQ tests are rescaled onto this standard scale. Standard scores are important for making comparisons across groups and data collection situations. They allow us to compare apples to oranges, that is, to compare data that otherwise would not be comparable. For example, perhaps we wanted to determine in which class a student was performing better. Let's say that Sue has an 85% in her Algebra course and a 90% in her English course. We cannot say that she is doing better in her English class just because it's the highest grade. Instead, we would need to rescale the scores onto the same scale so that they can be compared. The most commonly used standard score in statistics is the z-score, and is exactly what we would use here to answer this question.