Calculate 33rd percentile, mean 50, sd 10 Step by Step

Step by step

The question

Question: X is normal, with mean 50 and SD 10: which value has 33% below it?

μ = 50, σ = 10: x with 33% below it?

A value with 33 percent below it is also called the 33th percentile.

33% below it = the 33th percentile

1 Shade the area

The bell, with the mean 50 in the middle.

x is the cut-off we want.

33 percent is below x: we shade it, on the left of x.

2 Area to the left

The z-table works with the area to the left of z.

z-table: area to the left of z

33 percent is 0.33.

It is already on the left.

area to the left = 0.33

Careful!

Less than half the bell is on the left.

So x is below the mean, and z is negative.

Careful: less than half on the left: z is negative

3 Find it inside the table

Now we look for 0.3300 inside the table.

0.3300 is in the table: row −0.4, column .04.

So z = −0.44.

4 Back to x

Start at the mean, then go z SDs: x is the mean plus z times the SD.

x = μ + z × σ

−0.44 times 10 is −4.4.

50 minus 4.4 is 45.6.

x = 50 + (−0.44) × 10 = 50 − 4.4 = 45.6

5 Check

Let's check: the z of 45.6 is 45.6 minus 50, over 10.

That is −0.44.

z = 45.6 − 5010 = −0.44

The table at −0.44 gives 0.33: about 0.33 to the left.

So it is right!

table at z = −0.44: 0.33 ≈ 0.33

Let's look back.

First: shade the area.

Then: area to the left.

Then: find it inside the table.

Then: back to x.

Last: check.

Answer