1. A simple random sample of FICO credit rating scores is listed below:
714 751 664 789 818 779 698 836 753 834 693 802
a. Find range, variance, and standard deviation.
b. As of this writing, the mean FICO score was reported to be 678. Based on these results, is a FICO score of 500 unusual? Why or why not?
To find the range, variance, and standard deviation of the given FICO credit rating scores, you can follow these steps:
a. Range:
The range is the difference between the highest and lowest values in the dataset.
- Highest value: 836
- Lowest value: 664
- Range = Highest value - Lowest value = 836 - 664 = 172
b. Variance:
Variance measures the average squared deviation of each score from the mean. To calculate the variance, you will need to find the mean first.
- Mean:
To find the mean, sum up all the values and divide it by the number of values.
Mean = (714 + 751 + 664 + 789 + 818 + 779 + 698 + 836 + 753 + 834 + 693 + 802) / 12 = 7724 / 12 = 643.67 (rounded to two decimal places)
- Deviation:
For each score, subtract the mean from that score.
Deviation = Score - Mean
- Squared Deviation:
Square the deviation for each score.
Squared Deviation = (Score - Mean)^2
- Variance:
Variance is the average of the squared deviations.
Variance = Sum of Squared Deviations / (Number of Values - 1)
Variance = Sum of (Squared Deviations) / (12 - 1)
You can now calculate the variance using the values obtained above.
c. Standard Deviation:
The standard deviation is the square root of the variance.
Standard Deviation = √Variance
For part b, to determine if a FICO score of 500 is unusual, you can compare it with the mean and standard deviation.
- If a score is more than 2 standard deviations below or above the mean, it is considered unusual.
To calculate the zone of the mean:
Lower Zone = Mean - (2 * Standard Deviation)
Upper Zone = Mean + (2 * Standard Deviation)
If the score is below the lower zone or above the upper zone, it can be considered unusual.
Now, let's perform the calculations to find the range, variance, standard deviation, and determine if a score of 500 is unusual using the mean and standard deviation values.
1a. Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
Range = highest score - lowest
1b. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score. Make your decision from that.
I'll let you do the calculations.