Answer to Question #139723 in Statistics and Probability for Sohyun Ravena

Question

Instructions: Show your solutions,
Follow naming guidelines in the syllabus.
I. Using the table, find the following:
1. Mode, Median, Mode
2. Mean Deviation
3. Variance and Standard Deviation

72 44 80 40 44 60 80 43 55 62
73 49 58 48 80 53 52 47 51 80
46 61 41 74 78 56 62 55 75 51
64 40 79 65 76 66 46 60 67 52
76 49 44 47 45 68 57 56 59 68

Answer 1

Answer 1)

Mean $\bar x$ = Average of all numbers

= (72+44+80+40+44+60+80+43+55+62+73+49+58+48+80+53+52+47+51+80+46+61+41+

74+78+56+62+55+75+51+64+40+79+65+76+66+46+60+67+52+76+49+44+47+45+68+57+56+59+68) / 50

= 2954/50 = 59.08

Hence Mean $\bar x$ = 59.08

To find the mode we arrange the numbers in the ascending order

40, 40, 41, 43, 44, 44, 44, 45, 46, 46, 47, 47, 48, 49, 49, 51, 51, 52, 52, 53, 55, 55, 56, 56, 57, 58, 59, 60, 60, 61, 62, 62, 64, 65, 66, 67, 68, 68, 72, 73, 74, 75, 76, 76, 78, 79, 80, 80, 80, 80

The number 80 appears 4 times (which is the maximum), hence the mode = 80

Since the total no. of samples are even the median = $\frac {57+58}{2} = 57.5$

Answer 2)

Formula for mean deviation = $\frac {1}{n}\sum$_i=1 ⁵⁰ |x_i - $\bar x$|

where,

n = total number of sample

$\bar x =$ sample mean

Hence by substituting the values in the above formula we get,

Mean Deviation = $\frac{|72 - 59.08| + |44 - 59.08| + .\ .\ . + |59 - 59.08| + |68 - 59.08|}{50}$ = 534.32/50 = 10.6864.

Answer 3)

Variance = $\frac{\sum (x_i - \bar x)^2}{n}$ = $\frac{(72 - 59.08)^2 + (44 - 59.08)^2 + ... + (59 - 59.08)^2 + (68 - 59.08)^2}{50}$

$Variance = \frac{7745.68}{50} = 154.9136$

Standard Deviation = $\sqrt{Variance} = \sqrt{154.9136} = 12.446$