A set of data is normally distributed with a mean of 200 and standard deviation of 50.
• What would be the standard score for a score of 250?
• What percentage of scores is between 200 and 250?
• What would be the percentile rank for a score of 250?
1. Z = (score-mean)/SD
2. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Multiply proportion by 100 to get percentage.
3. Precentile rank is percentage ≤ the score. Use same table.
To find the answers to your questions, we can use the properties of the standard normal distribution.
1. Standard score (or z-score):
The standard score (or z-score) measures how many standard deviations a particular data point is from the mean. It is calculated using the formula: z = (x - μ) / σ
Given:
Mean (μ) = 200
Standard deviation (σ) = 50
Score (x) = 250
Substituting the values in the formula:
z = (250 - 200) / 50
z = 50 / 50
z = 1
Therefore, the standard score for a score of 250 is 1.
2. Percentage of scores between 200 and 250:
To find the percentage of scores between 200 and 250, we need to calculate the area under the normal distribution curve between these two scores. We can use a standard normal table or a statistical calculator to find this area.
Using a standard normal table, we find that the area to the left of the score 200 is 0.5, and the area to the left of the score 250 is 0.8413. Therefore, the area between 200 and 250 can be calculated as: 0.8413 - 0.5 = 0.3413.
To convert this area into a percentage, multiply by 100:
0.3413 * 100 = 34.13%
Therefore, approximately 34.13% of scores are between 200 and 250.
3. Percentile rank for a score of 250:
The percentile rank represents the percentage of scores that fall at or below a particular score.
To find the percentile rank for a score of 250, we need to calculate the area to the left of the score on the normal distribution curve.
Using a standard normal table, we find that the area to the left of the score 250 is 0.8413. Therefore, the percentile rank for a score of 250 can be calculated as: 0.8413 * 100 = 84.13%.
Therefore, the percentile rank for a score of 250 is approximately 84.13%.
To answer these questions, we need to understand the concept of standard scores, z-scores, and percentiles.
1. The standard score, also known as the z-score, measures how many standard deviations a particular score is from the mean. To calculate the standard score, you can use the formula:
z = (X - μ) / σ
Where:
X = Score
μ = Mean
σ = Standard Deviation
Plugging in the given values, we can calculate the standard score for a score of 250:
z = (250 - 200) / 50
z = 50 / 50
z = 1
Therefore, the standard score for a score of 250 is 1.
2. To find the percentage of scores between 200 and 250, we can utilize the standard normal distribution table. This table provides the area (percentage) under the normal curve up to a certain z-score. By finding the z-scores for both 200 and 250, we can calculate the percentage between them.
For 200:
z1 = (200 - 200) / 50
z1 = 0
For 250:
z2 = (250 - 200) / 50
z2 = 1
Looking up the z-scores in the standard normal distribution table, we find that the percentage for z = 0 is 0.5000 and the percentage for z = 1 is 0.8413.
The percentage of scores between 200 and 250 can be calculated by subtracting the smaller percentage from the larger percentage:
Percentage = 0.8413 - 0.5000
Percentage = 0.3413
Therefore, approximately 34.13% of scores are between 200 and 250.
3. The percentile rank of a score represents the percentage of scores that are equal to or below that particular score.
To calculate the percentile rank for a score of 250, we need to find the percentage of scores less than or equal to 250.
Using the z-score formula, we can calculate the z-score for 250:
z = (250 - 200) / 50
z = 1
By looking up the z-score of 1 in the standard normal distribution table, we find the percentage to be 0.8413.
Since the percentile rank represents the percentage of scores equal to or below a particular score, we can say that the percentile rank of a score of 250 is 84.13%.
In summary:
- The standard score of a score of 250 is 1.
- Approximately 34.13% of scores are between 200 and 250.
- The percentile rank for a score of 250 is 84.13%.