Question:

Find the mean and variance for the data 6,7,10,12,13,4,8,12

Answer:

Mean:

Step 1: Add all of the data points together: 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72

Step 2: Divide the sum by the number of data points: 72 ÷ 8 = 9

Therefore, the mean is 9.

Variance:

Step 1: Calculate the mean: 6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72; 72 ÷ 8 = 9

Step 2: Subtract the mean from each data point and square the result: (6 - 9)² = 9, (7 - 9)² = 4, (10 - 9)² = 1, (12 - 9)² = 9, (13 - 9)² = 16, (4 - 9)² = 25, (8 - 9)² = 1, (12 - 9)² = 9

Step 3: Add all of the squared results together: 9 + 4 + 1 + 9 + 16 + 25 + 1 + 9 = 73

Step 4: Divide the sum of the squared results by the number of data points: 73 ÷ 8 = 9.125

Therefore, the variance is 9.125.

Question:

Find the mean and variance for the first n natural numbers

Answer:

Step 1: Calculate the mean of the first n natural numbers. Mean = (n+1)/2

Step 2: Calculate the variance of the first n natural numbers. Variance = (n^2 - 1)/12

Question:

Find the mean and variance for the data xi​ 6 10 14 18 24 28 30 fi​ 2 4 7 12 8 4 3

Answer:

Step 1: Find the sum of all the values of xi​: xi​ 6 10 14 18 24 28 30 fi​ 2 4 7 12 8 4 3

Sum of xi​ = 6 + 10 + 14 + 18 + 24 + 28 + 30 = 120

Step 2: Find the sum of all the values of fi​: fi​ 2 4 7 12 8 4 3

Sum of fi​ = 2 + 4 + 7 + 12 + 8 + 4 + 3 = 40

Step 3: Find the mean: Mean = (Sum of xi​) / (Sum of fi​) Mean = 120 / 40 Mean = 3

Step 4: Find the variance: Variance = [(xi​ - Mean)2 x fi​] / (Sum of fi​) Variance = [(6 - 3)2 x 2 + (10 - 3)2 x 4 + (14 - 3)2 x 7 + (18 - 3)2 x 12 + (24 - 3)2 x 8 + (28 - 3)2 x 4 + (30 - 3)2 x 3] / 40 Variance = [9 x 2 + 49 x 4 + 91 x 7 + 153 x 12 + 576 x 8 + 784 x 4 + 841 x 3] / 40 Variance = 11,068 / 40 Variance = 276.7

Question:

Find the mean and variance for the data xi​ 92 93 97 98 102 104 109 fi​ 3 2 3 2 6 3 3

Answer:

Step 1: Find the mean. Mean = (92 x 3 + 93 x 2 + 97 x 3 + 98 x 2 + 102 x 6 + 104 x 3 + 109 x 3)/(3 + 2 + 3 + 2 + 6 + 3 + 3) Mean = (2748 + 1862 + 2910 + 1956 + 6120 + 3120 + 3267)/20 Mean = 20,823/20 Mean = 1041.15

Step 2: Find the variance. Variance = [(92 - 1041.15)2 x 3 + (93 - 1041.15)2 x 2 + (97 - 1041.15)2 x 3 + (98 - 1041.15)2 x 2 + (102 - 1041.15)2 x 6 + (104 - 1041.15)2 x 3 + (109 - 1041.15)2 x 3]/20 Variance = [(−49.15)2 x 3 + (−48.15)2 x 2 + (−44.15)2 x 3 + (−43.15)2 x 2 + (−39.15)2 x 6 + (−37.15)2 x 3 + (−32.15)2 x 3]/20 Variance = [2411.0225 + 2304.0225 + 1957.0225 + 1858.0225 + 1529.0225 + 1382.0225 + 1026.0225]/20 Variance = 10,599.3/20 Variance = 529.965

Hence, the mean and variance for the given data are 1041.15 and 529.965 respectively.

Question:

The diameters of circles (in mm) drawn in a design are given below : Diameters No. of circles 33−36 15 37−40 17 41−44 21 45−48 22 49−52 25

Answer:

Step 1: Calculate the total number of circles in the design.

Total number of circles = 15 + 17 + 21 + 22 + 25 = 100

Step 2: Calculate the sum of all the diameters of the circles.

Sum of all the diameters = (33 + 36) x 15 + (37 + 40) x 17 + (41 + 44) x 21 + (45 + 48) x 22 + (49 + 52) x 25 = 10,455

Step 3: Calculate the average diameter of the circles.

Average diameter = 10,455 / 100 = 104.55 mm

Question:

Find the mean and standard deviation using short-cut method Xi​ 60 61 62 63 64 65 66 67 68 fi​ 2 1 12 29 25 12 10 4 5

Answer:

Step 1: Find the mean. Mean = (60 x 2) + (61 x 1) + (62 x 12) + (63 x 29) + (64 x 25) + (65 x 12) + (66 x 10) + (67 x 4) + (68 x 5) / (2 + 1 + 12 + 29 + 25 + 12 + 10 + 4 + 5)

Mean = 6,506 / 87

Mean = 74.76

Step 2: Find the standard deviation. Standard Deviation = √[(60-74.76)2 x 2 + (61-74.76)2 x 1 + (62-74.76)2 x 12 + (63-74.76)2 x 29 + (64-74.76)2 x 25 + (65-74.76)2 x 12 + (66-74.76)2 x 10 + (67-74.76)2 x 4 + (68-74.76)2 x 5] / (2 + 1 + 12 + 29 + 25 + 12 + 10 + 4 + 5)

Standard Deviation = √[(-14.76)2 x 2 + (-13.76)2 x 1 + (-12.76)2 x 12 + (-11.76)2 x 29 + (-10.76)2 x 25 + (-9.76)2 x 12 + (-8.76)2 x 10 + (-7.76)2 x 4 + (-6.76)2 x 5] / 87

Standard Deviation = 3.853

Question:

Find the mean and variance for the first 10 multiples of 3

Answer:

Step 1: List the first 10 multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Step 2: Calculate the mean: Mean = (3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30) / 10 Mean = 135 / 10 Mean = 13.5

Step 3: Calculate the variance: Variance = [(3 - 13.5)2 + (6 - 13.5)2 + (9 - 13.5)2 + (12 - 13.5)2 + (15 - 13.5)2 + (18 - 13.5)2 + (21 - 13.5)2 + (24 - 13.5)2 + (27 - 13.5)2 + (30 - 13.5)2] / 10 Variance = [(-10.5)2 + (-7.5)2 + (-4.5)2 + (-1.5)2 + (1.5)2 + (4.5)2 + (7.5)2 + (10.5)2 + (13.5)2 + (16.5)2] / 10 Variance = [110.25 + 56.25 + 20.25 + 2.25 + 2.25 + 20.25 + 56.25 + 110.25 + 182.25 + 272.25] / 10 Variance = 835 / 10 Variance = 83.5

Question:

Find the mean and variance for the following frequency distribution Classes 0-30 30-60 60-90 90-120 120-150 150-180 180-210 Frequencies 2 3 5 10 3 5 2

Answer:

Mean:

Add the midpoints of each class: 0 + 30 + 60 + 90 + 120 + 150 + 180 = 630

Divide by the total number of classes: 630 / 7 = 90

Mean = 90

Variance:

Step 1: Find the deviation of each class from the mean: 0-30: 30 - 90 = -60 30-60: 60 - 90 = -30 60-90: 90 - 90 = 0 90-120: 120 - 90 = 30 120-150: 150 - 90 = 60 150-180: 180 - 90 = 90 180-210: 210 - 90 = 120

Step 2: Square the deviations: (-60)2 = 3600 (-30)2 = 900 02 = 0 302 = 900 602 = 3600 902 = 8100 1202 = 14400

Step 3: Multiply the squared deviations by the frequencies: 3600 x 2 = 7200 900 x 3 = 2700 0 x 5 = 0 900 x 10 = 9000 3600 x 3 = 10800 8100 x 5 = 40500 14400 x 2 = 28800

Step 4: Add the products from Step 3: 7200 + 2700 + 0 + 9000 + 10800 + 40500 + 28800 = 104400

Step 5: Divide the sum from Step 4 by the total number of frequencies: 104400 / 25 = 4176

Variance = 4176

Question:

Find the mean and variance for the following frequency distrubution Classes 0-10 10-20 20-30 30-40 40-50 Frequencies 5 8 15 16 6

Answer:

Mean =

(05 + 108 + 2015 + 3016 + 40*6)/(5+8+15+16+6)

Mean = 20

Variance =

[(0-20)^25 + (10-20)^28 + (20-20)^215 + (30-20)^216 + (40-20)^2*6]/(5+8+15+16+6)

Variance = 200

Question:

Find the mean variance and standard deviation using short-cut method Height in cms No. of children 70-75 3 75-80 4 80-85 7 85-90 7 90-95 15 95-100 9 100-105 6 105-110 6 110-115 3

Answer:

Step 1: Calculate the Mid-point of each class:

Height (cms) No. of children Mid-point (x) 70-75 3 72.5 75-80 4 77.5 80-85 7 82.5 85-90 7 87.5 90-95 15 92.5 95-100 9 97.5 100-105 6 102.5 105-110 6 107.5 110-115 3 112.5

Step 2: Calculate the Frequency (f):

Height (cms) No. of children Mid-point (x) Frequency (f) 70-75 3 72.5 3 75-80 4 77.5 4 80-85 7 82.5 7 85-90 7 87.5 7 90-95 15 92.5 15 95-100 9 97.5 9 100-105 6 102.5 6 105-110 6 107.5 6 110-115 3 112.5 3

Step 3: Calculate the Sum of Mid-point (x) and Frequency (f):

Sum of x = 72.5 + 77.5 + 82.5 + 87.5 + 92.5 + 97.5 + 102.5 + 107.5 + 112.5 = 825

Sum of f = 3 + 4 + 7 + 7 + 15 + 9 + 6 + 6 + 3 = 60

Step 4: Calculate the Mean (x̅):

Mean (x̅) = 825/60 = 13.75

Step 5: Calculate the Deviation (d):

Deviation (d) = x - x̅

Step 6: Calculate the Variance (σ2):

Variance (σ2) = d2 x f

Step 7: Calculate the Standard Deviation (σ):

Standard Deviation (σ) = √σ2