For 3 and 4 find the range, variance, and standard deviation. Round to the nearest tenth.
3. Twelve students were given an arithmetic test and the times (in minutes) to complete it were
10, 9, 12, 11, 8, 15, 9, 7, 8, 6, 12, 10
4. The normal daily temperatures (in degrees Fahrenheit) in January for 10 selected cities are as follows.
50, 37, 29, 54, 30, 61, 47, 38, 34, 61
5. If a history test has a mean of 100 and a standard deviation of 10, find the corresponding z score for a test score of 115.
http://easycalculation.com/statistics/standard-deviation.php
5.
115 - 100 = 15
z score means how many standard deviations away from mean
so
15/10 = 1.5
A+ :)
To do the calculations yourself for 3 and 4:
Range = highest score - lowest
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
For 5, this is the equation used:
Z = (score-mean)/SD
To find the range, variance, and standard deviation for the given data set, we'll need to follow these steps:
1. Range: The range is the difference between the maximum and minimum values in a data set. To find the range, we'll subtract the minimum from the maximum.
For data set 3 (arithmetic test times):
Arrange the data in ascending order: 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 15.
Range = Maximum value - Minimum value = 15 - 6 = 9.
For data set 4 (daily temperatures):
Arrange the data in ascending order: 29, 30, 34, 37, 38, 47, 50, 54, 61, 61.
Range = Maximum value - Minimum value = 61 - 29 = 32.
2. Variance: Variance measures how spread out the data set is. The formula for variance is sum of squared deviations from the mean divided by the number of data points. To find the variance, follow these steps:
For data set 3:
Calculate the mean (average) of the data set: (10 + 9 + 12 + 11 + 8 + 15 + 9 + 7 + 8 + 6 + 12 + 10) / 12 = 9.
Calculate the squared deviation from the mean for each data point:
(10 - 9)^2 = 1, (9 - 9)^2 = 0, (12 - 9)^2 = 9, (11 - 9)^2 = 4, (8 - 9)^2 = 1, (15 - 9)^2 = 36, (9 - 9)^2 = 0, (7 - 9)^2 = 4, (8 - 9)^2 = 1, (6 - 9)^2 = 9, (12 - 9)^2 = 9, (10 - 9)^2 = 1.
Sum the squared deviations: 1 + 0 + 9 + 4 + 1 + 36 + 0 + 4 + 1 + 9 + 9 + 1 = 75.
Calculate the variance: 75 / 12 = 6.25.
For data set 4:
Calculate the mean of the data set: (50 + 37 + 29 + 54 + 30 + 61 + 47 + 38 + 34 + 61) / 10 = 43.1.
Calculate the squared deviation from the mean for each data point:
(50 - 43.1)^2 = 47.61, (37 - 43.1)^2 = 37.21, (29 - 43.1)^2 = 200.01, (54 - 43.1)^2 = 119.21, (30 - 43.1)^2 = 171.61, (61 - 43.1)^2 = 322.81, (47 - 43.1)^2 = 15.61, (38 - 43.1)^2 = 26.01, (34 - 43.1)^2 = 83.61, (61 - 43.1)^2 = 322.81.
Sum the squared deviations: 47.61 + 37.21 + 200.01 + 119.21 + 171.61 + 322.81 + 15.61 + 26.01 + 83.61 + 322.81 = 1,346.58.
Calculate the variance: 1,346.58 / 10 = 134.66.
3. Standard Deviation: The standard deviation is the square root of the variance. To find the standard deviation, take the square root of the variance.
For data set 3:
Standard deviation = √6.25 = 2.5
For data set 4:
Standard deviation = √134.66 ≈ 11.6
4. To find the z-score corresponding to a test score of 115, we'll use the formula: z = (score - mean) / standard deviation. Let's substitute the given values into the formula:
z = (115 - 100) / 10 = 1.5
Therefore, the z-score corresponding to a test score of 115 is 1.5.
Remember to round the values as mentioned in the question, so the final answers are:
- For data set 3: Range = 9, Variance = 6.3, Standard deviation = 2.5
- For data set 4: Range = 32, Variance = 134.7, Standard deviation = 11.6
- Z-score for a test score of 115 = 1.5