calculate the range, variance, standard deviation, and interquartile range for the data set given {72, 80, 80, 82, 88, 90, 96}
To calculate the range, variance, standard deviation, and interquartile range for the given data set {72, 80, 80, 82, 88, 90, 96}, follow these steps:
Step 1: Arrange the data set in ascending order: {72, 80, 80, 82, 88, 90, 96}
Step 2: Calculate the range:
Range = Maximum value - Minimum value
Range = 96 - 72
Range = 24
Step 3: Calculate the variance:
Variance measures how far each number in the set is from the mean and is calculated using the formula:
Variance = Sum of (each value in the set - mean) squared / Total number of values in the set
First, calculate the mean:
Mean = Sum of all values in the set / Total number of values in the set
Mean = (72 + 80 + 80 + 82 + 88 + 90 + 96) / 7
Mean = 588 / 7
Mean = 84
Next, calculate the sum of squared deviations:
Squared Deviations = (72 - 84)^2 + (80 - 84)^2 + (80 - 84)^2 + (82 - 84)^2 + (88 - 84)^2 + (90 - 84)^2 + (96 - 84)^2
Squared Deviations = (-12)^2 + (-4)^2 + (-4)^2 + (-2)^2 + (4)^2 + (6)^2 + (12)^2
Squared Deviations = 144 + 16 + 16 + 4 + 16 + 36 + 144
Squared Deviations = 376
Finally, calculate the variance:
Variance = Squared Deviations / Total number of values in the set
Variance = 376 / 7
Variance ≈ 53.71
Step 4: Calculate the standard deviation:
Standard Deviation = Square Root of Variance
Standard Deviation = Square Root of 53.71
Standard Deviation ≈ 7.33
Step 5: Calculate the interquartile range:
The interquartile range (IQR) measures the dispersion of the data set, particularly around the median.
First, calculate the median:
Median = Middle value of the data set
Median = (80 + 82) / 2
Median = 81
Next, calculate the lower quartile (Q1):
Lower Quartile (Q1) = Median of the lower half of the data set
Q1 = (72 + 80) / 2
Q1 = 76
Then, calculate the upper quartile (Q3):
Upper Quartile (Q3) = Median of the upper half of the data set
Q3 = (88 + 90) / 2
Q3 = 89
Finally, calculate the interquartile range (IQR):
IQR = Q3 - Q1
IQR = 89 - 76
IQR = 13
So, for the given data set {72, 80, 80, 82, 88, 90, 96}, the range is 24, the variance is approximately 53.71, the standard deviation is approximately 7.33, and the interquartile range is 13.
To calculate the range, variance, standard deviation, and interquartile range for the given data set {72, 80, 80, 82, 88, 90, 96}, follow these steps:
1. Range:
The range is the difference between the maximum and minimum values in the data set.
- Maximum value: 96
- Minimum value: 72
- Range = Maximum value - Minimum value = 96 - 72 = 24
2. Variance:
Variance measures the spread or dispersion of the data from the mean. It is calculated by finding the average of the squared differences from the mean.
- Mean (average):
Sum of all values / Number of values = (72 + 80 + 80 + 82 + 88 + 90 + 96) / 7 = 88
- Squared differences from the mean:
(72 - 88)^2, (80 - 88)^2, (80 - 88)^2, (82 - 88)^2, (88 - 88)^2, (90 - 88)^2, (96 - 88)^2
= 256, 64, 64, 36, 0, 4, 64
- Variance:
Sum of squared differences / Number of values = (256 + 64 + 64 + 36 + 0 + 4 + 64) / 7 = 48
3. Standard Deviation:
The standard deviation is the square root of the variance. It measures the average amount of variation or dispersion in the data set.
- Standard Deviation = √Variance = √48 ≈ 6.93
4. Interquartile Range (IQR):
The interquartile range is the range between the first quartile (Q1) and the third quartile (Q3). It represents the middle 50% of the data.
- To find the quartiles, first, arrange the data set in ascending order:
72, 80, 80, 82, 88, 90, 96
- Q1: First quartile (25th percentile):
- Find the position of Q1 using the formula: (n + 1) × (1st quartile / 100)
where n is the number of data points.
- (7 + 1) × (25 / 100) = 8 × 0.25 = 2 (rounded down)
- The 2nd value in the ordered data set is 80, which is Q1.
- Q3: Third quartile (75th percentile):
- Find the position of Q3 using the formula: (n + 1) × (3rd quartile / 100)
- (7 + 1) × (75 / 100) = 8 × 0.75 = 6 (rounded up)
- The 6th value in the ordered data set is 90, which is Q3.
- IQR = Q3 - Q1 = 90 - 80 = 10
Therefore, for the given data set:
- Range = 24
- Variance = 48
- Standard Deviation ≈ 6.93
- Interquartile Range = 10
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
Interquartile range = 75th percentile - 25th percentile
I'll let you do the calculations.