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High School Algebra I: Help and Review25 chapters | 292 lessons

Instructor:
*Michael Perekupka*

In this lesson, you will learn about the properties of the standard normal distribution and why it is so important to the field of statistics. A real life example will be used to illustrate this value.

Big news: your midterm grades are in. You scored an 84 in Calculus, a 93 in Spanish, and a 79 in AP physics. News in hand, you run downstairs to share your grades. The time comes for you to blurt out your results, and suddenly a barrage of questions begin running through your head - which score will I share first? Which am I most proud of?

Surely the 93 is bestâ€¦ Right? 93 > 84 > 79, simple math. But, the Spanish test was easier. Plus, your friend just sent you a text message saying she scored a 61 in Calculus. AP Physics has challenged you all year, and your last test score was in 50s, so the 79 was a major improvement. How will you decide which score was your biggest triumph?

What you need is a way to put these scores on the same scale. The solution to all of your problems lays within the **Standard Normal Distribution**, a distribution that assigns scores based on performance relative to how others performed in the same population.

In order to make the proper comparisons in the example above, let's assume (perhaps a bit unreasonably) that all midterms scores were normally distributed. However, there are infinitely many normal distributions in the world, each one centered around its own average, with its own standard deviation. For instance, Calculus midterms had an average of 73 with a standard deviation of 8, Spanish midterms had an average of 87 with a standard deviation of 6, and Physics midterms had an average of 70, with a standard deviation of 15. Recall that standard deviation is a measure of spread; it is an average distance of data points away from the mean.

Thus, comparing across distributions is akin to comparing apples to oranges. To make an informed decision, we will covert all scores to the same scale. We will **standardize** the scores, converting them to the standard normal distribution, with a single mean and a single standard deviation.

The standard normal distribution, commonly referred to the Z-distribution, is a special case of a normal distribution with the following properties:

- It has a mean of zero.
- It has a standard deviation of one.
- Areas under the density curve can be found using a
**standard normal table**

A score on the standard normal distribution is called a **Z-Score**. It should be interpreted as the number of standard deviations a data point is above or below the mean. A positive Z-Score indicates that a point is above the average, and a negative Z-Score indicates a score below the average.

The standard normal distribution follows the **Empirical Rule** (68-95-99.7 Rule), stating that 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% of data falls within three standard deviations of the mean.

A Z-Score, also referred to as a standardized score, is the result of converting from one of the infinitely many normal distributions to the standard normal distribution. Z-scores allow us to make comparisons and calculate probabilities easily when working with normal distributions. To make the conversion - or to standardize - the following formula is used:

Z = (Raw score - Average) / Standard Deviation

The raw score is the point of interest, and the average and standard deviation refer to the values from its original normal distribution.

Let's look at our example from the beginning of the lesson and figure out which score you should share first by standardizing all of your scores and converting them to Z-Scores on the Standard Normal Distribution. Once they are all on the same scale - with an average of 0, and a standard deviation of 1, we can determine which is the most impressive. The work to accomplish this task is shown below:

Calculus: Z = (84 - 73) / 8 = 1.38

Spanish: Z = (93 - 87) / 6 = 1.00

AP Phys: Z = (79 - 70) / 15 = 0.60

(Note: Z-Scores are typically rounded to two decimal points.)

Let's examine the results. Your 84 in calculus equates to a Z-Score of 1.38 on the Standard Normal Distribution. Your 93 in Spanish equates to a 1.00, and lastly your 79 in AP Physics gives a Z-Score of 0.60. To reiterate, this means that you scored 1.38 standard deviations above the mean in calculus, 1 standard deviation above the mean in Spanish, and 0.60 standard deviations above the mean in AP Physics. Now there is no debate; your score in Calculus is the most impressive relative to how others performed on the same test - share that first!

The standard normal distribution allows us to make comparisons across the infinitely many normal distributions that exist in the world. A score on the standard normal distribution is called a Z-Score, and is interpreted as the number of standard deviations a data point falls above or below the mean. The standard normal distribution has the specific properties of having an average of zero and a standard deviation of one and evaluates performances relative to others in the same population.

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High School Algebra I: Help and Review25 chapters | 292 lessons

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