Given the data set:
2;3;3;5;8;9;12;13;14
1. Determine the mean
2. Determine the mode
3. Determine the median
4. Determine the upper quartile
5. Determine the standard deviation
1. Mean:
(2 + 3 + 3 + 5 + 8 + 9 + 12 + 13 + 14) / 9 = 69 / 9 = 7.67
2. Mode:
There is no mode in this data set as each number appears only once.
3. Median:
Since there are 9 numbers in the data set, the median is the middle number when the numbers are arranged in ascending order.
The numbers in ascending order are:
2;3;3;5;8;9;12;13;14
The median is the 5th number, which is 8.
4. Upper Quartile:
To find the upper quartile, we first need to find the median of the upper half of the data set. In this case, the upper half is:
9;12;13;14
Since there are 4 numbers in the upper half, the upper quartile is the median of those numbers. The numbers in ascending order are:
9;12;13;14
The median of these numbers is the average of the 2nd and 3rd number, which is (12 + 13)/2 = 12.5
5. Standard Deviation:
The formula for standard deviation is the square root of the variance. First, we need to find the variance of the data set:
Variance = [Σ(xi - x̄)²] / (n - 1)
Where Σ is the sum of, xi is each data point, x̄ is the mean, and n is the number of data points.
Calculating the variance:
Variance = [(2-7.67)² + (3-7.67)² + (3-7.67)² + (5-7.67)² + (8-7.67)² + (9-7.67)² + (12-7.67)² + (13-7.67)² + (14-7.67)²] / 8
Variance = [31.7089 + 19.1089 + 19.1089 + 7.6089 + 0.7744 + 3.7089 + 18.6729 + 29.7225 + 39.8721] / 8
Variance = 170.8874 / 8
Variance = 21.3609
Now, calculating the standard deviation:
Standard Deviation = sqrt(Variance)
Standard Deviation = sqrt(21.3609)
Standard Deviation = 4.62
So, the standard deviation of the data set is 4.62.