AP Stats normal distribution z-scores

What is a z-score?

A z-score measures how many standard deviations a data point is from the mean of a distribution.

Calculate the z-score: X = 70, μ = 65, σ = 5.

Use the formula: $\displaystyle z = \\frac{X - \mu}{\sigma}$. Here, $\displaystyle z = \\frac{70 - 65}{5} = 1$.

True or False: A z-score of 2 means the value is above average.

True. A z-score of 2 indicates the value is 2 standard deviations above the mean.

Interpret a z-score of -1.5.

The value is 1.5 standard deviations below the mean, indicating it's relatively low.

Find the z-score for a test score of 80, mean 75, SD 10.

Calculate: $\displaystyle z = \\frac{80 - 75}{10} = 0.5$. This score is half a standard deviation above the mean.

Difference between z-scores and percentiles?

Z-scores indicate standard deviations from the mean; percentiles indicate the percentage of values below a certain point.

Fill in the blank: A z-score of 0 indicates a value _____.

equal to the mean.

Calculate the z-score for: X = 50, μ = 55, σ = 5.

Use the formula: $\displaystyle z = \\frac{50 - 55}{5} = -1$. The value is below average.

How is a negative z-score interpreted?

It indicates the value is below the mean.

Example of z-score in real life?

In test scores, a z-score of 1.2 shows a student scored better than average.

What does a z-score of 3 signify?

It signifies a value that is 3 standard deviations above the mean, indicating it's quite high.

Calculate z-score: X = 90, μ = 85, σ = 4.

Calculate: $\displaystyle z = \\frac{90 - 85}{4} = 1.25$. This shows the score is above average.

True or False: A z-score of -2.5 is typical.

False. A z-score of -2.5 indicates a very low value, quite rare in a normal distribution.

What can z-scores tell us about a paycheck?

A z-score can show how a paycheck compares to the average for a job position; e.g., +1 indicates a higher salary.

Z-score interpretation example: Z = 0.75.

This indicates the value is 0.75 standard deviations above the mean.

Calculate the z-score for: X = 30, μ = 25, σ = 5.

Z = \\frac{30 - 25}{5} = 1. This means the score is 1 standard deviation above the mean, indicating it is relatively high compared to the average.

Normal distribution shape

Symmetrical bell curve centered around the mean, with tails extending infinitely.

What percentage of data falls within one standard deviation?

About 68% of data falls within one standard deviation of the mean in a normal distribution.

Empirical Rule components

68-95-99.7 Rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ.

True or False: The mean, median, and mode are equal.

True. In a normal distribution, the mean, median, and mode coincide at the center.

Standard deviation effect

Larger standard deviation: Wider curve, more spread out data points.

What area corresponds to z = 0?

The area to the left of z = 0 is 0.5, or 50% of the data.

Fill in the blank: The total area under a normal distribution curve is _____

1 or 100%.

Comparison: Normal vs. Skewed distribution

Normal: Symmetrical; Skewed: Asymmetrical, tail on one side.

Z-score interpretation

A z-score indicates how many standard deviations a value is from the mean.

What does a negative z-score indicate?

A value below the mean in a normal distribution.

How does the area change with z-scores?

Higher z-scores correspond to smaller areas in the tail; lower z-scores cover larger areas.

True or False: All normal distributions have the same standard deviation.

False. Different normal distributions can have different standard deviations.

Z-score for scores above average

Positive z-scores indicate scores above the mean in a normal distribution.

Standard normal distribution mean and SD

Mean = 0, Standard Deviation = 1.

What percentage is within 2 standard deviations?

About 95% of data falls within two standard deviations of the mean.

Effect of increasing sample size

Larger sample size leads to better approximation of normal distribution (Central Limit Theorem).

Fill in the blank: The tails of the normal distribution approach _____

The horizontal axis but never touch it.

What is the area between z = -1 and z = 1?

Approximately 68% of the total area under the curve.

Use case for normal distribution

Modeling heights of people, test scores, or measurement errors.

What area under the curve corresponds to z = 1?

Approximately 0.8413, meaning 84.13% of data falls below this z-score.

Calculate the z-score for a test score of 85.

Given a mean of 75 and a standard deviation of 10: $\displaystyle z = \\frac{85 - 75}{10} = 1.0$

A company reports an average salary of $\displaystyle 60,000 with a standard deviation of$5,000. What is the z-score for a salary of $70,000?

Using the formula: $\displaystyle z = \\frac{70,000 - 60,000}{5,000} = 2.0$

True or False: A z-score of -2 indicates a value significantly below the mean.

True - A z-score of -2 means the value is two standard deviations below the mean.

A student scores in the 90th percentile. What does the z-score indicate?

The z-score for the 90th percentile is approximately 1.28, meaning the score is above the average.

How does a higher z-score affect your percentile rank?

Higher z-scores correspond to higher percentiles, indicating a better performance relative to the mean.

If a store's average sale is $\displaystyle 200 with a standard deviation of$50, what is the z-score for a $300 sale?

$\displaystyle z = \\frac{300 - 200}{50} = 2.0$; this sale is two standard deviations above average.

Fill in the blank: A z-score of 0 means the value is _______.

equal to the mean.

How would you interpret a z-score of 1.5 for a monthly rent of $1,500?

If mean rent is $\displaystyle 1,200 with a standard deviation of$200, the z-score indicates it's 1.5 standard deviations above average.

Calculate the z-score for a weight of 180 pounds if the mean is 150 and SD is 30.

$\displaystyle z = \\frac{180 - 150}{30} = 1.0$; signifies above-average weight.

True or False: Lower z-scores always indicate better performance.

False - Lower z-scores indicate performance further from the mean, possibly worse.

What is the z-score for a score of 65 in a class where the mean is 70 and SD is 5?

$\displaystyle z = \\frac{65 - 70}{5} = -1.0$; signifies below average.

A product has a lifespan mean of 5 years with a SD of 1 year. What is the z-score for a lifespan of 6 years?

$\displaystyle z = \\frac{6 - 5}{1} = 1.0$; above average lifespan.

In a data set, which z-score indicates a relative outlier?

A z-score beyond |3| is typically considered an outlier.

Calculate the z-score for a car's fuel efficiency of 30 MPG if the mean is 25 MPG and SD is 3 MPG.

$\displaystyle z = \\frac{30 - 25}{3} \approx 1.67$; above average efficiency.

How do you interpret a negative z-score in a business context?

A negative z-score indicates a performance or measurement below the average.

If the average exam score is 78 with a SD of 10, what z-score corresponds to a score of 88?

$\displaystyle z = \\frac{88 - 78}{10} = 1.0$; signifies above-average performance.

Fill in the blank: A z-score of 2.5 indicates a value that is _______ standard deviations above the mean.

2.5

Calculate the z-score for a game score of 95 with a mean of 85 and SD of 5.

$\displaystyle z = \\frac{95 - 85}{5} = 2.0$; significantly above average.

Comparing two employees: One has a z-score of 1.5 and the other -0.5. Who performed better?

The employee with a z-score of 1.5 performed better; higher z-scores indicate better performance.

True or False: A z-score of 0.5 is a sign of excellent performance.

False - A z-score of 0.5 indicates slightly above average, not excellent.

True or False: A z-score of -1 indicates a below-average performance.

True. A z-score of -1 means the value is 1 standard deviation below the mean, indicating below-average performance.

Calculate the z-score for a paycheck of $\displaystyle 2,500 if the average is$3,000 and SD is $400.

$z = \frac{X - ext{mean}}{ ext{SD}} = \frac{2500 - 3000}{400} = -1.25. This indicates the paycheck is below average.

Find the probability of z < 1.5.

Using z-tables or calculators, P(Z < 1.5) ≈ 0.9332.

True or False: A z-score of 2.0 means 2 standard deviations above the mean.

True – A z-score indicates how many standard deviations a value is from the mean.

Find the percentile for z = -1.0.

P(Z < -1.0) ≈ 0.1587, meaning it's in the 15.87th percentile.

If the mean is $\displaystyle 50 and SD is$10, what is P(X < $60)?

Calculate z: $\displaystyle z = \frac{60-50}{10} = 1.0$. Then, P(Z < 1.0) ≈ 0.8413.

What is z for the 95th percentile?

z ≈ 1.645. This means 95% of values lie below this z-score.

Fill in the blank: P(Z > 0) = ____

0.5 – Since the normal distribution is symmetric about the mean.

Calculate P(z < -2.5).

Using tables or calculators, P(Z < -2.5) ≈ 0.0062.

What does a z-score of -1.5 indicate?

It indicates the value is 1.5 standard deviations below the mean.

Find the probability of z > 2.

P(Z > 2) = 1 - P(Z < 2) ≈ 1 - 0.9772 = 0.0228.

A score of 70 has a z-score of 0.5. Find mean and SD.

If $\displaystyle X = 70$, then $\displaystyle X = ext{mean} + 0.5 imes ext{SD}$.

True or False: A higher z-score always means a higher probability.

False – Higher z-scores reflect lower probabilities in the upper tail.

What is the probability of z between -1 and 1?

P(-1 < Z < 1) = P(Z < 1) - P(Z < -1) ≈ 0.8413 - 0.1587 = 0.6826.

Calculate the z-score for a test score of 85.

If mean = 80 and SD = 5, $\displaystyle z = \frac{85 - 80}{5} = 1.0$.

Find percentile for z = 1.96.

P(Z < 1.96) ≈ 0.9750, indicating the 97.5th percentile.

What z-score corresponds to the 5th percentile?

z ≈ -1.645. This indicates a low score compared to the population.

If a z-score is 3, what’s the probability of being less?

P(Z < 3) ≈ 0.9987. Very few scores exceed this z-score.

Find the probability between z = -2 and z = 2.

P(-2 < Z < 2) = P(Z < 2) - P(Z < -2) ≈ 0.9772 - 0.0228 = 0.9544.

What does a z-score of 0 represent?

It represents a value equal to the mean.

Calculate P(Z < -0.5).

P(Z < -0.5) ≈ 0.3085.

Find probability for z > 1.5.

P(Z > 1.5) = 1 - P(Z < 1.5) ≈ 1 - 0.9332 = 0.0668.

Find the probability of z < -1.25.

Using the z-table, P(Z < -1.25) ≈ 0.1056. This means roughly 10.56% of the data falls below this z-score.

A z-score of 1.5 corresponds to which percentile?

Using the z-table, z = 1.5 corresponds to the 93.32nd percentile. This means 93.32% of the data is below this z-score.