In these set of discrete, ungrouped data: 60, 62, 63, 65, 59, 59, 58, 60, 62, 62, 63, 63, 59, 61, 60, 64, 61, 63. Find (calculate) the mean, mode, median, range, interquartile range, variance, standard deviation, and mean deviation from the given discrete, ungrouped data.
To find the mean, add up all the numbers and divide by the total number of values:
60 + 62 + 63 + 65 + 59 + 59 + 58 + 60 + 62 + 62 + 63 + 63 + 59 + 61 + 60 + 64 + 61 + 63 = 1,104
Mean = 1,104 ÷ 18 = 61.33
To find the mode, identify the value(s) that appear most frequently. In this case, the mode is 63, as it appears 4 times.
To find the median, first arrange the data in ascending order:
58, 59, 59, 59, 60, 60, 60, 61, 61, 62, 62, 63, 63, 63, 64, 65
Since there are 18 values, the median will be (18+1)/2 = 9.5th value.
The 9th value is 61, and the 10th value is 62. Thus, the median is (61 + 62)/2 = 61.5.
To find the range, subtract the lowest value from the highest value:
65 - 58 = 7
To find the interquartile range, first find the first quartile (Q1). Since 25% of the data is below Q1, Q1 will be the 25th percentile. The 5th value is 60, and the 6th value is 61. Thus, Q1 = (60 + 61) / 2 = 60.5.
To find the third quartile (Q3), find the 75th percentile, which is the 75th value. The 14th value is 63, and the 15th value is 64. Thus, Q3 = (63 + 64) / 2 = 63.5.
The interquartile range is Q3 - Q1 = 63.5 - 60.5 = 3.
To find the variance, first calculate the squared differences between each data point and the mean:
(60 - 61.33)^2 + (62 - 61.33)^2 + (63 - 61.33)^2 + (65 - 61.33)^2 + (59 - 61.33)^2 + (59 - 61.33)^2 + (58 - 61.33)^2 + (60 - 61.33)^2 + (62 - 61.33)^2 + (62 - 61.33)^2 + (63 - 61.33)^2 + (63 - 61.33)^2 + (59 - 61.33)^2 + (61 - 61.33)^2 + (60 - 61.33)^2 + (64 - 61.33)^2 + (61 - 61.33)^2 + (63 - 61.33)^2
Simplifying this expression (sum of squared differences) gives:
200.96 + 0.448 + 0.098 + 13.728 + 2.528 + 2.528 + 8.368 + 0.448 + 0.448 + 0.448 + 0.098 + 0.098 + 2.528 + 0.088 + 0.448 + 7.328 + 0.088 + 0.098 = 249.08
To find the variance, divide this result by (n-1), where n is the total number of values:
Variance = 249.08 / (18-1) = 14.65
To find the standard deviation, take the square root of the variance:
Standard Deviation = √14.65 ≈ 3.828
To find the mean deviation, first calculate the absolute differences between each data point and the mean:
|60 - 61.33| + |62 - 61.33| + |63 - 61.33| + |65 - 61.33| + |59 - 61.33| + |59 - 61.33| + |58 - 61.33| + |60 - 61.33| + |62 - 61.33| + |62 - 61.33| + |63 - 61.33| + |63 - 61.33| + |59 - 61.33| + |61 - 61.33| + |60 - 61.33| + |64 - 61.33| + |61 - 61.33| + |63 - 61.33|
Simplifying this expression (sum of absolute differences) gives:
1.33 + 0.67 + 1.67 + 3.67 + 2.33 + 2.33 + 3.33 + 1.33 + 0.67 + 0.67 + 1.67 + 1.67 + 2.33 + 0.33 + 0.67 + 2.67 + 0.67 + 1.67 = 27.33
To find the mean deviation, divide this result by the total number of values:
Mean Deviation = 27.33 / 18 ≈ 1.52
So, the mean is 61.33, the mode is 63, the median is 61.5, the range is 7, the interquartile range is 3, the variance is 14.65, the standard deviation is 3.828, and the mean deviation is 1.52.