Given the data below \( 4,7,5,9,8,3,6 \) find: (i) The mean (i) The mide. (iv) The medrain (w) The firit scand and thire moment (a) The rigion (b) Assumed, mean of 7

Let's solve the problem step by step. ### Given Data The data set is: \( 4, 7, 5, 9, 8, 3, 6 \) ### (i) Mean The mean is calculated by summing all the values and dividing by the number of values. \[ \text{Mean} = \frac{\sum \text{data}}{n} \] Where \( n \) is the number of data points. ### (ii) Mode The mode is the value that appears most frequently in the data set. ### (iii) Median The median is the middle value when the data set is ordered. If there is an even number of observations, the median is the average of the two middle numbers. ### (iv) First Standard and Third Moment - The first standard moment is the mean. - The third moment about the mean is calculated using the formula: \[ \text{Third Moment} = \frac{1}{n} \sum (x_i - \text{mean})^3 \] ### (v) Range The range is calculated as: \[ \text{Range} = \text{Maximum} - \text{Minimum} \] ### (vi) Assumed Mean of 7 To find the assumed mean, we can use the formula for the mean deviation about an assumed mean: \[ \text{Mean Deviation} = \frac{\sum |x_i - A|}{n} \] Where \( A \) is the assumed mean. Now, let's perform the calculations for each of these steps. 1. Calculate the mean. 2. Determine the mode. 3. Find the median. 4. Calculate the first standard moment and third moment. 5. Calculate the range. 6. Calculate the mean deviation about the assumed mean of 7. Let's proceed with the calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(4+7+5+9+8+3+6\right)}{7}\) - step1: Remove the parentheses: \(\frac{4+7+5+9+8+3+6}{7}\) - step2: Add the numbers: \(\frac{42}{7}\) - step3: Reduce the numbers: \(\frac{6}{1}\) - step4: Calculate: \(6\) Calculate or simplify the expression \( (4-mean)^3 + (7-mean)^3 + (5-mean)^3 + (9-mean)^3 + (8-mean)^3 + (3-mean)^3 + (6-mean)^3 \). Simplify the expression by following steps: - step0: Solution: \(\left(4-mean\right)^{3}+\left(7-mean\right)^{3}+\left(5-mean\right)^{3}+\left(9-mean\right)^{3}+\left(8-mean\right)^{3}+\left(3-mean\right)^{3}+\left(6-mean\right)^{3}\) - step1: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-mean\right)^{3}+\left(5-mean\right)^{3}+\left(9-mean\right)^{3}+\left(8-mean\right)^{3}+\left(3-mean\right)^{3}+\left(6-mean\right)^{3}\) - step2: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-eman\right)^{3}+\left(5-mean\right)^{3}+\left(9-mean\right)^{3}+\left(8-mean\right)^{3}+\left(3-mean\right)^{3}+\left(6-mean\right)^{3}\) - step3: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-eman\right)^{3}+\left(5-eman\right)^{3}+\left(9-mean\right)^{3}+\left(8-mean\right)^{3}+\left(3-mean\right)^{3}+\left(6-mean\right)^{3}\) - step4: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-eman\right)^{3}+\left(5-eman\right)^{3}+\left(9-eman\right)^{3}+\left(8-mean\right)^{3}+\left(3-mean\right)^{3}+\left(6-mean\right)^{3}\) - step5: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-eman\right)^{3}+\left(5-eman\right)^{3}+\left(9-eman\right)^{3}+\left(8-eman\right)^{3}+\left(3-mean\right)^{3}+\left(6-mean\right)^{3}\) - step6: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-eman\right)^{3}+\left(5-eman\right)^{3}+\left(9-eman\right)^{3}+\left(8-eman\right)^{3}+\left(3-eman\right)^{3}+\left(6-mean\right)^{3}\) - step7: Reorder the terms: \(\left(4-eman\right)^{3}+\left(7-eman\right)^{3}+\left(5-eman\right)^{3}+\left(9-eman\right)^{3}+\left(8-eman\right)^{3}+\left(3-eman\right)^{3}+\left(6-eman\right)^{3}\) - step8: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+\left(7-eman\right)^{3}+\left(5-eman\right)^{3}+\left(9-eman\right)^{3}+\left(8-eman\right)^{3}+\left(3-eman\right)^{3}+\left(6-eman\right)^{3}\) - step9: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+343-147eman+21e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+\left(5-eman\right)^{3}+\left(9-eman\right)^{3}+\left(8-eman\right)^{3}+\left(3-eman\right)^{3}+\left(6-eman\right)^{3}\) - step10: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+343-147eman+21e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+125-75eman+15e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+\left(9-eman\right)^{3}+\left(8-eman\right)^{3}+\left(3-eman\right)^{3}+\left(6-eman\right)^{3}\) - step11: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+343-147eman+21e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+125-75eman+15e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+729-243eman+27e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+\left(8-eman\right)^{3}+\left(3-eman\right)^{3}+\left(6-eman\right)^{3}\) - step12: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+343-147eman+21e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+125-75eman+15e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+729-243eman+27e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+512-192eman+24e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+\left(3-eman\right)^{3}+\left(6-eman\right)^{3}\) - step13: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+343-147eman+21e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+125-75eman+15e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+729-243eman+27e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+512-192eman+24e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+27-27eman+9e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+\left(6-eman\right)^{3}\) - step14: Expand the expression: \(64-48eman+12e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+343-147eman+21e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+125-75eman+15e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+729-243eman+27e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+512-192eman+24e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+27-27eman+9e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}+216-108eman+18e^{2}m^{2}a^{2}n^{2}-e^{3}m^{3}a^{3}n^{3}\) - step15: Add the numbers: \(2016-840eman+126e^{2}m^{2}a^{2}n^{2}-7e^{3}m^{3}a^{3}n^{3}\) It seems there were some issues with the calculations. Let's go through each step manually to ensure we get the correct results. ### (i) Mean The mean is calculated as follows: \[ \text{Mean} = \frac{4 + 7 + 5 + 9 + 8 + 3 + 6}{7} = \frac{42}{7} = 6 \] ### (ii) Mode The mode is the number that appears most frequently. In this data set, all numbers appear only once, so there is no mode. ### (iii) Median To find the median, we first sort the data: Sorted data: \( 3, 4, 5, 6, 7, 8, 9 \) Since there are 7 numbers (an odd count), the median is the middle number: \[ \text{Median} = 6 \] ### (iv) First Standard Moment and Third Moment - The first standard moment is the mean, which we found to be \( 6 \). - To calculate the third moment about the mean, we use the formula: \[ \text{Third Moment} = \frac{1}{n} \sum (x_i - \text{mean})^3 \] Calculating each term: - For \( 4 \): \( (4 - 6)^3 = (-2)^3 = -8 \) - For \( 7 \): \( (7 - 6)^3 = (1)^3 = 1 \) - For \( 5 \): \( (5 - 6)^3 = (-1)^3 = -1 \) - For \( 9 \): \( (9 - 6)^3 = (3)^3 = 27 \) - For \( 8 \): \( (8 - 6)^3 = (2)^3 = 8 \) - For \( 3 \): \( (3 - 6)^3 = (-3)^3 = -27 \) - For \( 6 \): \( (6 - 6)^3 = (0)^3 = 0 \) Now summing these values: \[ \text{Sum} = -8 + 1 - 1 + 27 + 8 - 27 + 0 = 0 \] Thus, the third moment is: \[ \text{Third Moment} = \frac{0}{7} = 0 \] ### (v) Range The range is calculated as: \[ \text{Range} = \text{Maximum} - \text{Minimum} = 9 - 3 = 6 \] ### (vi) Assumed Mean of 7 To calculate the mean deviation about the assumed mean of 7: \[ \text{Mean Deviation} = \frac{\sum |x_i - 7|}{n} \] Calculating each term: - For \( 4 \): \( |4 - 7| = 3 \) - For \( 7 \): \( |7 - 7| = 0 \) - For \( 5 \): \( |5 - 7| = 2 \) - For \( 9 \): \( |9 - 7| = 2 \) - For \( 8 \): \( |8 - 7| = 1 \) - For \( 3 \): \( |3 - 7| = 4 \) - For \( 6 \): \( |6 - 7| = 1 \) Now summing these values: \[ \text{Sum} = 3 + 0 + 2 + 2 + 1 + 4 + 1 = 13 \] Thus, the mean deviation is: \[ \text{Mean Deviation} = \frac{13}{7} \approx 1.857 \] ### Summary of Results - Mean: \( 6 \) - Mode: No mode - Median: \( 6 \) - First Standard Moment: \( 6 \) - Third Moment: \( 0 \) - Range: \( 6 \) - Mean Deviation about assumed mean of \( 7 \): \( \approx 1.857 \) If you have any further questions or need additional calculations, feel free to ask!