Describe the measures of central tendency and measures of dispersion for the following data set:
A = {8,1,2,9,3,2,8,1,2}
I will not find the values for you, but here is how you can get them.
For the mean, add all the scores and divided by the number of scores.
μ = ΣX/N
The mode is the most frequently occurring score.
For the median, first you must arrange the scores in order of their value. The middle-most score is the median.
The range is the highest score minus the lowest.
This should help you with the standard deviation:
http://www.easycalculation.com/statistics/learn-standard-deviation.php
I hope this helps. Thanks for asking.
To find the measures of central tendency and measures of dispersion for the given data set A = {8, 1, 2, 9, 3, 2, 8, 1, 2}, we will calculate the mean, median, mode, range, variance, and standard deviation.
1. Mean (average):
To find the mean, add up all the values in the data set and then divide by the total number of values.
Sum of all values = 8 + 1 + 2 + 9 + 3 + 2 + 8 + 1 + 2 = 36
Number of values = 9
Mean = Sum of all values / Number of values
Mean = 36 / 9
Mean = 4
2. Median:
To find the median, first, arrange the data set in ascending order.
1, 1, 2, 2, 2, 3, 8, 8, 9
Since the data set has an odd number of values, the median will be the middle value.
Median = 2
3. Mode:
The mode is the value that appears most frequently in the data set.
In this data set, the mode is 2 because it appears three times, more than any other number.
4. Range:
The range is the difference between the largest and smallest values in the data set.
Range = Largest value - Smallest value
Range = 9 - 1
Range = 8
5. Variance:
Variance is a measure of how much the values in the data set vary around the mean.
To calculate the variance, we need to find the squared differences between each value and the mean, then take the average of these squared differences.
Calculate the squared differences:
(8 - 4)^2 = 16
(1 - 4)^2 = 9
(2 - 4)^2 = 4
(9 - 4)^2 = 25
(3 - 4)^2 = 1
(2 - 4)^2 = 4
(8 - 4)^2 = 16
(1 - 4)^2 = 9
(2 - 4)^2 = 4
Sum of squared differences = 16 + 9 + 4 + 25 + 1 + 4 + 16 + 9 + 4 = 88
Variance = Sum of squared differences / Number of values
Variance = 88 / 9
Variance = 9.7778
6. Standard Deviation:
Standard deviation is the square root of the variance, and it measures how spread out the values are from the mean.
Standard Deviation = Square root of the variance
Standard Deviation = Square root of 9.7778
Standard Deviation = 3.1257 (rounded to four decimal places)
So, the measures of central tendency for the data set A are:
Mean = 4
Median = 2
Mode = 2
And the measures of dispersion for the data set A are:
Range = 8
Variance = 9.7778
Standard Deviation = 3.1257
To describe the measures of central tendency and measures of dispersion for the given data set A = {8,1,2,9,3,2,8,1,2}, we need to find the mean, median, mode, range, variance, and standard deviation.
Measures of Central Tendency:
1. Mean: The mean is calculated by adding up all the values in the data set and dividing the sum by the number of values. In this case,
Mean = (8 + 1 + 2 + 9 + 3 + 2 + 8 + 1 + 2) / 9
= 36 / 9
= 4
2. Median: The median is the middle value of the data set when arranged in ascending order. In this case, the data set in ascending order is {1, 1, 2, 2, 2, 3, 8, 8, 9}. The middle value is 2, so the median is 2.
3. Mode: The mode is the value(s) that appear most frequently in the data set. In this case, the value 2 appears the most (3 times), so the mode is 2.
Measures of Dispersion:
1. Range: The range is calculated by subtracting the minimum value from the maximum value in the data set. In this case, the minimum value is 1 and the maximum value is 9, so the range is 9 - 1 = 8.
2. Variance: The variance measures how spread out the data set is from the mean. It is calculated by taking the average of the squared differences between each value and the mean.
Differences from the mean: (8-4)^2, (1-4)^2, (2-4)^2, (9-4)^2, (3-4)^2, (2-4)^2, (8-4)^2, (1-4)^2, (2-4)^2
Variance = (16 + 9 + 4 + 25 + 1 + 4 + 16 + 9 + 4) / 9
= 88 / 9
≈ 9.778
3. Standard Deviation: The standard deviation is the square root of the variance. In this case, the standard deviation ≈ √9.778 ≈ 3.13.
So, the measures of central tendency for the data set A = {8,1,2,9,3,2,8,1,2} are mean = 4, median = 2, mode = 2, and the measures of dispersion are range = 8, variance ≈ 9.778, and standard deviation ≈ 3.13.