Percentiles
A student scored at the 70th percentile on a test. What does this mean?
The student scored higher than 70% of the other students.
What does a z‑score of 0 mean?
The value is exactly at the mean.
Adding a constant to every value affects which: mean, SD, or both?
Mean changes, SD stays the same.
About what percent of data falls within 1 SD of the mean in a normal distribution?
68%
What are the mean and SD of the standard normal distribution?
Mean=0
SD=1
A baby’s weight is at the 40th percentile. What percent of babies weigh more?
60% weigh more.
A z‑score is positive. What does that tell you?
The value is above the mean.
Multiplying every value by 3 affects which: mean, SD, or both?
Both mean and SD are multiplied by 3.
About what percent falls within 2 SD?
95%
What area is to the left of z = 0 on the standard normal curve?
0.5
A runner’s time is at the 20th percentile in a race. What does this tell you about how the runner performed compared to others?
The runner preformed better than 20 percent of runners and worse than 80% of them
A test score is 85, the mean is 75, and the SD is 5. What is the z‑score?
z=(85−75)/5=2.
If you convert inches to centimeters (multiply by 2.54), what happens to z‑scores?
They stay the same.
About what percent falls within 3 SD?
99.7%
Which value is larger: the area to the left of z = 1 or the area to the left of z = –1?
Area to the left of z = 1 (about 0.8413).
What is the difference between the 75th percentile and Q3?
Q3 is the 75th percentile.
A z‑score of –1.5 means what in context?
The value is 1.5 standard deviations below the mean.
A distribution has mean 50 and SD 10. After subtracting 5 from every value, what are the new mean and SD?
Mean = 45, SD = 10.
In a normal distribution with mean 100 and SD 10, what range covers 95% of data?
80-120
Find the approximate area between z = –1 and z = 2.
Area left of 2= 0.9772
Area left of -1= 0.1587
Difference= 0.8185
A student is at the 90th percentile in height and the 60th percentile in weight. What does this tell you?
They are taller than 90% of students but heavier than only 60%, meaning they are tall but not unusually heavy for their height.
Two students have z‑scores of 1.2 (math) and 1.8 (reading). Who performed better relative to their group?
A distribution has mean 20 and SD 4. You transform values using y=3x+2. What are the new mean and SD?
3(20)+2=62
SD = 3(4)=12.
A normal distribution has mean 60 and SD 8. What percent of data is between 52 and 76?
52 is 1 SD below, 76 is 2 SD above 68% + (95%−68%)/2 = 81.5%
A value has a z‑score of 1.75. Use the standard normal distribution to estimate the percentile.
Area left of 1.75= 0.9599, about 96th percentile