The proportion of data below a value is the area under the curve to its left. The whole area is 1.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Percentiles
A percentile is that proportion as a percentage. The 90th percentile has 90% of the data below it.
Familiar landmarks
z = 0 is the 50th percentile. z = 1 is about the 84th, since half of the middle 68% sits above the mean plus the other 16% below.
Proportions above
To find the proportion above a value, subtract the proportion below from 1.
Between two values
Subtract the smaller left-hand area from the larger one. What remains is the strip in between.
Working backwards
Given a percentile, find its z, then reverse the formula: value = mean + z × standard deviation.
Probability is area under the curve
The proportion of values in an interval is the area under the normal curve over it. The total area is 1, which is why probabilities and proportions are the same computation here.
Reading a table or a calculator
Standard tables give the area to the left of a z-score. Areas to the right come from subtracting from 1, and areas between from subtracting one from the other. Sketching first prevents the wrong subtraction.
From percentile back to value
Given a percentile, find the z-score and then reverse the standardisation to recover the original value. This is how cut-off scores for the top 10% of anything are computed.
Always sketch
Draw the curve, mark the mean and the value, and shade the region you want. Nearly every error in normal probability problems is shading or subtracting the wrong region, and the sketch is what catches it.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Elena Find a Percentile Value
Scores are normal with mean 500 and standard deviation 100. Elena wants the 84th percentile.
- Step 1
The 84th percentile corresponds to z = 1, since 50% sits below the mean and 34% more within one standard deviation.
Step 4: Your Turn
Practice makes it stick.
The Median
Problem 1 of 2
What percentile does z = 0 correspond to?
The Value
Problem 2 of 2
Mean 500, sd 100, z = 1. What is the value?
Areas and Percentiles
1 of 8
z = 0. Percentile?
2 of 8
z = 1. Approximate percentile?
3 of 8
z = −1. Approximate percentile?
4 of 8
60% lies below a value. What proportion lies above, as a decimal?
5 of 8
Mean 200, sd 20, z = 2. What is the value?
6 of 8
Areas below two values are 0.84 and 0.16. What proportion lies between them?
7 of 8
Match each z-score with its approximate percentile.
Tap a card on the left to start.
8 of 8
Mean 70, sd 5, z = −2. What is the value?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Mean 100, sd 15, z = 1. What is the value?
Question 2 of 2
What does the area to the left of a value represent?
What You Learned
- Area to the left of a value gives the proportion below it.
- z = 0, 1 and −1 mark roughly the 50th, 84th and 16th percentiles.
- Reverse the z formula to turn a percentile back into a value.