Z-scores

The grades on a math midterm are normally distributed with mean of 67% and a standard deviation of 2.5. Greg scored a 70%. What is his z-score?

1.2

In the problem above, what percentage of the class scored below Greg? http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf

88.49%

What is a z-score?

A z-score represents the number of standard deviations away from the mean any given data point is in a normal distribution.

What does a z-score mean?

A z-score tells you how many standard deviations above or below the mean your data point it. A z-score of 1 is 1 standard deviation above the mean. A score of 2 is 2 standard deviations above the mean. A score of -1.8 is -1.8 standard deviations below the mean. A z-score of 0 is equal to the mean (exactly average).

You take the SAT and score 1100. The mean score for the SAT is 1026 and the standard deviation is 209. How well did you score on the test compared to the average test taker?

Step 1: Write your X-value into the z-score equation. For this sample question the X-value is your SAT score, 1100. Z=(1100-μ)/σ Step 2: Put the mean, μ, into the z-score equation. Z=(1100-1026)/σ Step 3: Write the standard deviation, σ into the z-score equation. Z=(1100-1026)/209 Step 4: Calculate the answer using a calculator: (1100 - 1026) / 209 = .354. This means that your score was .354 std devs above the mean.

Richard grows prize winning pumpkins. He grows a pumpkin which weighs 450 pounds and enters it into a contest. The average weight of pumpkins in the contest is 320 pounds with a standard deviation of 75 pounds. What percentage of pumpkins weigh more than Richard's pumpkin?

The z-score is 1.73, so the percentage of people with pumpkins weighing less than Richard's is 95.82%. This means that only 4.18% of pumpkins weighed more than his.

How is a z-score used in real life?

You can use the z-table and the normal distribution graph to give you a visual about how a z-score is higher or lower than average. The z-score in the center of the curve is zero. The z-scores to the right of the mean are positive and the z-scores to the left of the mean are negative. If you look up the score in the z-table, you can tell what percentage of the population is above or below your score.

z = x - μ / σ

z=(data point - mean)/standard deviation