Normal Distribution & Z Scores

A raw score (X)

the original observation or value before it has been transformed into a Z Score

What do you need in order to calculate a Z Score

the value that is given of a particular point on the distribution + mean + Standard deviation of distribution

Suppose we have a normal distribution with a mean of 100 and a standard deviation of 10. One value in that distribution is 120. How many Standard Deviations above the mean is that value?

z = (120 - 100) / 10 = 2

In a normal distribution, what portion of the values do you expect to less than one standard deviation away from the mean? 28% 68% 50% It depends on the distribution

68%

How to estimate the SD of a histogram

Make sure that the graph is unimodal, symmetric, bell-shaped (a normal distribution) 1. Look at where the mean is 2. locate the points at which the graphs decrease into the tail(lowest point) 3. Calculate the value (mean - smallest point) 4. divide that value by 3 because there are 3 standard deviations in between the mean and the lowest point

Normal Distribution - Why does it look like that?

Many variables tend to cluster around some average value with most values being a little above or a below average (middle where it peaks is the average, the left side is below average and the right side is above average)

Z = 0.6, what percentage of the distribution is above that value?

26%

What percentage of the normal distribution falls below a Z score of -1.8? 6% 2% 3% 12%

3 %

Z= -1.8, what percentage of the distribution is below that value?

3%

How much does this data fall within 1 Standard Deviation above the mean?

34% (this means that if the data distribution had 100 scores in it, we would expect roughly 34 (34% of the distribution) to be 1 SD above the mean)

Z = +1 , what percentage of the distribution is below that value?

84%

Z = -1.5 , what percentage of the distribution is above that value?

91%

What portion of the area under the normal curve falls between a Z score of +1 and a Z score of +2 About 2% About 34% About 14% About 10% None of these is correct

About 14%

In a normal distribution, what portion of the values do you expect to be more than two standard deviations above the mean? About 5% About 14% About 2% About 1%

About 2%

Which of the following measurements would you expect to be roughly normally distributed? The amount of raing that falls in Bloomington each year The weights of adult guinea pigs The number of cheeseburgers sold at McDonald's each day All of these

All of these

If a variable comes from a population, the the variable is represented with a ________ letter.

Greek

Suppose that you knew that the amount of rain that we received in Bloomington in 2017 corresponded to a a Z score of -2.5 in the distribution of yearly rainfall over Bloomington's history.What can you tell me about the amount of rain in Bloomington in 2017? It was about average It was far above average It was far below average it was below average it was above average

It was far below average ( a Z Score of -2.5 means that it is -2.5 SD away from the mean, that is pretty close to the end of the tail which means that it is extremely far away from the mean, so far below)

If a variable comes from a sample, the the variable is represented with a ________ letter.

Roman (English)

Sample Stats and Population Parameters

Sample Mean: Variance: SD: Population Mean: Variance: SD:

How much of the curve (the total distribution) is above the mean value?

Since the graph is normal and symmetric, 50%

Which of these is the most precise definition of a Z score? a measure of how many standard deviations a score is above or below the mean a measure of how extreme a value is a measure of how far a score is from the mean a measure of a score's variance relative to other scores a measure of where a score falls in the population relative to the mean value

a measure of how many standard deviations a score is above or below the mean

Z Score

a measure of how many standard deviations you are away from the norm (average or mean) Where a particular score falls within a distribution With the z score you can communicate if a particular value is above or below the mean and how "extreme" that score is

How much does this data fall within 2 Standard Deviations above the mean?

14% (this means that if the data distribution had 100 scores in it, we would expect roughly 14 (14% of the distribution) to be 1 to 2 SD above the mean)

How much does this data fall within 3 Standard Deviations above the mean?

2% (this means that if the data distribution had 100 scores in it, we would expect roughly 2 (2% of the distribution) to be 1 to 2 to 3 SD above the mean)

Suppose you knew that among the distribution of all college students, your IQ was associated with a Z score of 2.1. Where does that put you among your fellow college students? Way below average A little below average Right around average A little above average Well above average

Well above average

Raw Score (X) equation

X (raw score) = Z * SD + Mean

Z Score Equation

Z = X - Mean / SD

The average college student hiccups 17 times per month with a standard deviation of 4 hics. In case you are wondering, yes, I completely made those values up. Anyway, suppose that I asked a randomly selected IU student to count their hiccups for the past month and they reported 17 hics. What is the Z score for that result? -2 -0.5 0 0.5 2

0

Suppose that you were working with a set of scores that were normally distributed with a mean of 50 and a standard deviation of 10. One score in that distribution is a 54. What is the Z score associated with that raw score? -0.4 -0.1 0 0.1 0.4

0.4 PIC

Suppose you knew that amount of wool produced by an adult alpaca each year (each shear?) is normally distributed with a mean of 7 lbs and a standard deviation of 1.5 lbs. When you sheared your pet alpaca, you harvested 8 lbs. of wool. What is the Z score for that amount of wool? -1 -0.667 0 0.667 1.0

0.667

The number of cups of coffee I drink per day is approximately normally distributed. I drink an average of 4 cups, with a standard deviation of 1.5.Yesterday morning I had 6 cups of coffee. What Z score corresponds to that value? 0 1.33 3 1.5 -1

1.33

How much of the values are less than 1 SD above or below the mean?

68%

Suppose that the gas station near your house started running ads that the Z score for their gas price was Z = -0.05 compared to the city's other gas stations. Besides concluding that they've hired the worst. marketing agency. ever. What else do you know? That their gas prices are way, way below average That their gas prices are substantially below average That their gas prices are just barely below average That their gas prices are just a little above average That their gas prices are well above average

That their gas prices are just barely below average

Normal Distribution or Normal Curve

Unimodal, symmetric, bell-shaped

When someone talks about the normal distribution they are referring to: any unimodal and negatively skewed distribution any unimodal and positively skewed distribution any distribution that is unimodal and symmetric a particular, but common, type of unimodal and negatively skewed distribution a particular, but common, type of unimodal and positively skewed distribution a particular, but common, type of unimodal and symmetric distriubtion

a particular, but common, type of unimodal and symmetric distriubtion

Suppose we have a normal distribution with a mean of 100 and a standard deviation of 10. One value in that distribution is 85. How many Standard Deviations above the mean is that value?

z = (85 - 100) / 10 = -1.5

Suppose you knew that the individual cheese consumption of Americans was normally distributed with a mean of 44 lbs per year and a standard deviation of 4 lbs (more cheese examples? Really?!?). Suppose that one American ate 32 lbs. of cheese last year. What is the Z score associated with that cheese consumption? -3 -1 0 1 3

-3