(1) Giken the datakiow, \( 4,7,5,9,8,3,6 \) find: (i) The mean (ii) The mote (iii) The median (iv) The fuostmenent (v) Second moment (vi) The ongin (vii) Arsumed, mean of 7

We are given the seven‐value data set   4, 7, 5, 9, 8, 3, 6 and asked to find several “moments” and measures. (Note that the spellings are non‐standard, e.g. “mote” means mode, “fuostmenent” means first moment (about the origin), “ongin” is taken to be the central moment (variance), and “Arsumed, mean of 7” means the estimated mean using 7 as an assumed value.) Let’s work through each part. ───────────────────────────── Step 1. Preliminary Calculations a. It is often helpful to know the sample size and sum.  • Number of data, n = 7.  • Sum of the data = 4 + 7 + 5 + 9 + 8 + 3 + 6 = 42. ───────────────────────────── (i) The Mean The mean is the sum divided by the number of observations.  Mean = 42⁄7 = 6 ───────────────────────────── (ii) The Mote (Mode) The mode is the value(s) that occur most frequently. Here, every number appears exactly once. Thus, there is no unique mode (or you may say “no mode”).  Mode = None ───────────────────────────── (iii) The Median The median is the middle number when the data are arranged in order. First, sort the data:   3, 4, 5, 6, 7, 8, 9 Since there are 7 values, the median is the 4th one.  Median = 6 ───────────────────────────── (iv) The Fuostmenent (First Moment about the Origin) The “first moment about the origin” is essentially the mean. In other words, it is  First moment = (1/n) × (sum of data) = 42/7 = 6 ───────────────────────────── (v) Second Moment The “second moment about the origin” is the average of the squares of the data. We calculate each square then average them.  Calculate squares:   4² = 16, 7² = 49, 5² = 25, 9² = 81, 8² = 64, 3² = 9, 6² = 36  Sum of squares = 16 + 49 + 25 + 81 + 64 + 9 + 36 = 280  Second moment = 280⁄7 = 40 ───────────────────────────── (vi) The Ongin The wording “the ongin” is not standard, but in this context it is reasonable to interpret it as the central moment (the second central moment), which is the variance. (Remember that  Variance = (1/n) × [Σ(x – mean)²].) We already have the mean = 6, so find the squared deviations:  For x = 4: (4 – 6)² = 4  For x = 7: (7 – 6)² = 1  For x = 5: (5 – 6)² = 1  For x = 9: (9 – 6)² = 9  For x = 8: (8 – 6)² = 4  For x = 3: (3 – 6)² = 9  For x = 6: (6 – 6)² = 0  Sum of squared deviations = 4 + 1 + 1 + 9 + 4 + 9 + 0 = 28  Variance = 28⁄7 = 4 So, the “ongin” (central second moment) = 4 ───────────────────────────── (vii) Arsumed, Mean of 7 (Assumed Mean Method) Here we use an assumed value, A = 7. The idea is to write each observation as  x = A + d   (with d = deviation from 7) Then  Mean = A + (sum of deviations)/n Calculate the deviations:  For 4: d = 4 – 7 = –3  For 7: d = 7 – 7 = 0  For 5: d = 5 – 7 = –2  For 9: d = 9 – 7 = 2  For 8: d = 8 – 7 = 1  For 3: d = 3 – 7 = –4  For 6: d = 6 – 7 = –1 Sum of deviations = (–3) + 0 + (–2) + 2 + 1 + (–4) + (–1) = –7 Now,  Assumed mean = 7 + (–7)/7 = 7 – 1 = 6 ───────────────────────────── Summary of Answers 1. Mean = 6 2. Mode = None (each value occurs once) 3. Median = 6 4. First moment (about the origin) = 6 5. Second moment (about the origin) = 40 6. “Ongin” (variance, the second central moment) = 4 7. Assumed mean (using 7 as the assumed value) = 6 These are the results for the given data set.