Tiêu đề Descriptive Stat & Probability.pdf ✅

Variable: Is a parameter which is changing with outcome for and observed phenomena. Data : The particular value of the variable is data-value. Population: The collection of all data is called population. Sample: A subset of population is called sample. e.g. 1; Suppose a dice is thrown 45 times and outcomes are 6,1,6,2,1,6,6,4,1,5,5,5,1,2,6,6,4,2,3,2,4,1,4,4,6,4,3,6,4,3,2,6,4,3,4,3, 3,5,4,6,3,3,3,6,5. ...(A) It can be arranged as (A1) x = number 1 2 3 4 5 6 frequency 5 5 9 10 5 11 Cumulative freq. (≤) 5 10 19 29 34 45 Cumulative freq. (≥) 45 40 35 26 16 11 (A) Scattered data , & (A1) grouped data yielding the table called frequency distribution e.g, 2; Consider the temperature of a city we have 71.1, 71.4, 71.9, 72.8, 75.9, 76.6, 76.9, 78.6, 80.7, 81.6, 81.8, 83.0, 84.0, 86.2, 87.8, 87.9, 88.8, 88.9, 89.4, 91.9, 92.3, 94.1, 94.4, 94.4, 94.6, 94.7, 95.0, 96.0, 96.8, 99.2, 101.0, 101.7, 103.0, 106.0, 107.5 ...(B) It can be represented in form of class say 70-75, 75-80, 80-85, 85-90, 90-95, 95-100, 100-105, 105-110. The mid value of an interval is taken as class value. Any data falling on class boundaries to be assigned to higher class Data can be represented as Class bound 70-75 75-80 80-85 85-90 90-95 95-100 100- 105 105- 110 Class value 72.5 77.5 82.5 87.5 92.5 97.5 102.5 107.5 frequency 4 4 5 6 7 4 3 2 Cumulative freq. 4 8 13 19 26 30 33 35 Joining midpoint of the tips of the rectangle in the histogram. The polygon is closed on left and right by joining 67.5 112.5 and in this case sum of the area of rectangle equals to area bounded by frequency polygon and x − axis.
Mean The mean (Arithmetic mean) of sample values x1, x2, ... xn is given as sample mean (x̅) = x1 + x2 + ⋯ +xn n = ∑ xi n i=1 n or = f1x1 + f2x2 + ⋯ +fkxk f1 + f2 + ⋯ +fk = ∑ fixi k i=1 ∑ fi k i=1 Mean is called average value. e.g. 3; the mean of e.g. 1 is 5∗1+5∗2+9∗3+10∗4+5∗5+11∗6 45 = 173 45 = 3.84 Mid range The mid range of a sorted sample x1, x2, ... xn is the average of the smallest and the largest value , mid = x1 + xn 2 In case (A) mid = 1+6 2 = 3.5 & in case (B) mid = 72.5+107.5 2 = 90.0 Median (x̃) If the sample data x1, x2, ... xn is sorted in increasing order the median of the data is given as x̃ = { xk+1 if n = 2k + 1 xk+xk+1 2 if n = 2k e.g. 4; for 3,3,5,7,8 x̃ = 5 & for 1,2,5,5,7,8,8,9 x̃ = 5+7 2 = 6 In case of (A1) x̃ = 3 (23rd term), and in case of (B) since data is grouped into the class, we can find median in two ways. Since the number of data is 35 hence median is 18th term. Using the cumulative frequency it is the second term of class 85-90, thus i. Simply x̃ = 87.5 that is the class value. ii. Linearly interpolate in the class x̃ = 85 + 5 6 ∗ 5 = 89.166 0 4 4 5 6 7 4 3 2 0 0 2 4 6 8 Frequency Class Value Histogram 4 4 5 6 7 4 3 2 0 2 4 6 8 frequency polygon