Given the datalielow, \( 47,5,9,8,3,6 \) find: (i) The mean (ii) The mote (iii) The mediai (iv) The frostmement (v) Second moment (vi) The orgin. (Vii) Assumed, mean of 7

We are given the data set:   47, 5, 9, 8, 3, 6 Some of the terms in the question are written in a non‐standard way but we will interpret them as follows: 1. “Mean” – the arithmetic average. 2. “Mote” – almost certainly means “mode,” the most frequently occurring value. 3. “Mediai” – means “median,” the middle value when the data are sorted. 4. “Frostmement” – is interpreted as “first moment” (about the origin), which is defined as the arithmetic mean. 5. “Second moment” – the average of the squares of the data values, i.e. (Σx²)/n. 6. “The orgin” – is most likely “origin” (i.e. the point 0 where moments are measured). 7. “Assumed, mean of 7” – means that if one uses an assumed mean of 7 (a common technique in mental arithmetic) one should arrive at the same true mean. We now compute each one step‐by‐step. ────────────────────────────── Step (i): The Mean The mean is given by   Mean = (47 + 5 + 9 + 8 + 3 + 6) / 6 First, sum the numbers:   47 + 5 = 52   52 + 9 = 61   61 + 8 = 69   69 + 3 = 72   72 + 6 = 78 Thus,   Mean = 78 / 6 = 13 ────────────────────────────── Step (ii): The Mode The mode is the value that appears most frequently. In our data set, each number appears exactly once. Therefore, we say that there is no mode (or “no unique mode”). ────────────────────────────── Step (iii): The Median To find the median, first sort the data in ascending order:   3, 5, 6, 8, 9, 47 Since there are 6 numbers (an even count), the median is the average of the third and fourth numbers. The third number is 6 and the fourth number is 8. Thus,   Median = (6 + 8) / 2 = 7 ────────────────────────────── Step (iv): The First Moment The “first moment about the origin” is defined as   μ′₁ = (Σ x) / n This is just the mean, which we already found to be 13. ────────────────────────────── Step (v): The Second Moment The second moment about the origin is defined as   μ′₂ = (Σ x²) / n First, compute each square:   47² = 2209   5² = 25   9² = 81   8² = 64   3² = 9   6² = 36 Now sum these squares:   Total = 2209 + 25 = 2234   2234 + 81 = 2315   2315 + 64 = 2379   2379 + 9 = 2388   2388 + 36 = 2424 Divide by the number of values (6):   Second moment = 2424 / 6 = 404 ────────────────────────────── Step (vi): The Origin The term “origin” typically refers to the point 0 when measuring moments. In this context, the first and second moments were taken “about the origin” (0). ────────────────────────────── Step (vii): Using an Assumed Mean of 7 Sometimes an assumed mean (A) is used to simplify calculations. Let A = 7. Then the deviations (xᵢ – A) are calculated. Compute the deviations:   For 47: 47 − 7 = 40   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 sum these deviations:   40 + (–2) + 2 + 1 + (–4) + (–1) = 40 − 2 + 2 + 1 − 4 − 1 = 36 To get the correction to the assumed mean, divide the sum of deviations by n:   Correction = 36 / 6 = 6 Then the true mean is:   Mean = Assumed mean + Correction = 7 + 6 = 13 ────────────────────────────── Summary of Answers: (i) Mean = 13 (ii) Mode = No mode (each value occurs only once) (iii) Median = 7 (iv) First moment (about the origin) = 13 (v) Second moment (about the origin) = 404 (vi) The origin is 0 (i.e. moments are measured about 0) (vii) Using an assumed mean of 7, we also obtain the true mean as 13 Each of these results follows directly from the definitions and computations given above.