A college student takes a standardized test and scores a 163. If the mean is 155 and the standard deviation is 7, what is the student’s percentile rank? (Assume a normal distribution.)
Well, let's clown around with some statistics! To find the student's percentile rank, we need to determine how their score compares to others.
First, we calculate the z-score, which tells us how many standard deviations the student's score is from the mean. Using the formula:
z = (x - μ) / σ
Where:
x = student's score
μ = mean
σ = standard deviation
Plugging in the values from your question, we get:
z = (163 - 155) / 7
z ≈ 1.14
Now, we need to find the percentile rank associated with this z-score. We can use a standard normal distribution table or a calculator, but since I'm a silly clown bot, I'll just give you an estimate.
Based on a rough approximation, a z-score of 1.14 corresponds to around the 87th percentile. So we can say that the student's percentile rank is approximately 87%.
Remember, though, this is just a fun approximation. For precise calculations, it's best to use a proper statistical tool.
To find the student's percentile rank, we can use the standard normal distribution table.
First, we need to calculate the z-score for the student's score. The z-score measures how many standard deviations above or below the mean the student's score is.
The z-score formula is: z = (x - μ) / σ
Where:
x = student's score (163)
μ = mean (155)
σ = standard deviation (7)
Calculating the z-score:
z = (163 - 155) / 7
z = 8 / 7
z ≈ 1.143
Next, we can use the z-score to find the percentile rank from the standard normal distribution table. The percentile rank represents the percentage of scores that fall below a given z-score.
Looking up the z-score of 1.143 in the table, we find that the percentile rank is approximately 87%.
Therefore, the student's percentile rank is approximately 87%.
To determine the student's percentile rank, we need to find the area to the left of the student's score on the distribution curve. Here's how we can do that:
Step 1: Calculate the Z-score.
The Z-score represents the number of standard deviations the student's score is from the mean. It can be calculated using the formula:
Z = (X - μ) / σ
Where:
X = Student's score (163)
μ = Mean (155)
σ = Standard deviation (7)
Substituting the values into the formula:
Z = (163 - 155) / 7
Z = 8 / 7
Z = 1.14 (rounded to two decimal places)
Step 2: Find the percentile rank using the Z-score.
To find the percentile rank, we need to consult a Z-table or use a calculator.
Looking up the Z-score of 1.14 in the Z-table or using a calculator, we find that the area to the left of this Z-score is approximately 0.8729.
Step 3: Convert the area to percentile rank.
To convert the area to a percentile rank, we multiply it by 100.
Percentile rank = 0.8729 * 100
Percentile rank ≈ 87.29
Therefore, the student's percentile rank is approximately 87.29. This means the student scored as well as or higher than approximately 87.29% of the other test-takers.
If you plug in your numbers here
http://davidmlane.com/hyperstat/z_table.html
You will see that it is the 87th %ile.