(1) Giken the data below, \( 4,7,5,9,8,3,6 \) Find: (i) The mean (ii) The mote (IIi) The medrai (iv) The fustmenent (v) Second moment (vi) The ongin. (ViI) Assumed, megn of 7

To find the mean, mode, median, first moment, second moment, and origin for the data set \( 4, 7, 5, 9, 8, 3, 6 \), let’s break it down: (i) Mean: The mean is calculated by adding all the numbers together and dividing by the total count. Here, \( (4 + 7 + 5 + 9 + 8 + 3 + 6) / 7 = 6 \). (ii) Mode: The mode is the value that appears most frequently. In this set, all numbers appear once. Hence, there is no mode. (iii) Median: To find the median, we must arrange the numbers in ascending order: \( 3, 4, 5, 6, 7, 8, 9 \). The middle number is \( 6 \). (iv) First moment: This typically refers to the mean itself, which we've found to be \( 6 \). (v) Second moment: The second moment about the mean is found by squaring the difference of each number from the mean, summing these squares, and dividing by the number of observations. Here it’s \( [ (4-6)^2 + (7-6)^2 + (5-6)^2 + (9-6)^2 + (8-6)^2 + (3-6)^2 + (6-6)^2 ]/7 = 4 \). (vi) Origin: The origin in this context usually implies the mean around which our data is centered, which is \( 6 \). (vii) Assuming, mean of \( 7 \): If we assume the mean is 7, we’d need to adjust the data set or reconsider its values, as the current calculations show it’s 6. However, the numbers could be tossed around to suit analytical models predicting various distributions. So, in summary, we can burst some numbers, but they are dancing around that mean of 6! 🍀📊