Answered: he diameters of the ZnO spherical…

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.4: Distributions Of Data

Problem 7PPS

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Question

The diameters of the ZnO spherical grains consisting of ZnC₂O₄ (zinc oxalate) starting material were determined by scanning electron microscope (SEM) and the following results were obtained in (nm) units in the sample examined 87, 74, 72, 93, 69, 75, 61, 63, 64, 91, 63, 57, 65, 62 57.
a) Find the arithmetic mean and median values of the distribution?
b) Find the standard deviation of the grain diameters and the coefficient of variation values?

Definition Definition Middle value of a data set. The median divides a data set into two halves, and it also called the 50th percentile. The median is much less affected by outliers and skewed data than the mean. If the number of elements in a dataset is odd, then the middlemost element of the data arranged in ascending or descending order is the median. If the number of elements in the dataset is even, the average of the two central elements of the arranged data is the median of the set. For example, if a dataset has five items—12, 13, 21, 27, 31—the median is the third item in ascending order, or 21. If a dataset has six items—12, 13, 21, 27, 31, 33—the median is the average of the third (21) and fourth (27) items. It is calculated as follows: (21 + 27) / 2 = 24.

Expert Solution

Step 1

Arithmetic mean:

$M e a n = \frac{s u m o f v a l u e s}{N u m b e r o f v a l u e s} = \frac{87 + 74 + 72 + . . . . . + 62 + 57}{15} = 70.2$

Median:

Here, the sample size is 15, which is odd number.

Hence, median is the middle value of the ascending ordered data.

Ascending ordered data:

57
57
61
62
63
63
64
65
69
72
74
75
87
91
93

Middle value is 65.

Median=65

Step 2

Standard deviation:

x	(x-mean)^2
87	282.24
74	14.44
72	3.24
93	519.84
69	1.44
75	23.04
61	84.64
63	51.84
64	38.44
91	432.64
63	51.84
57	174.24
65	27.04
62	67.24
57	174.24
	$\sum {(X - m e a n)}^{2} = 1946.4$

$S \tan d a r d d e v i a t i o n = \sqrt{\frac{\sum {(X - m e a n)}^{2}}{n - 1}} = \sqrt{\frac{1946.4}{15 - 1}} = 11.7910$

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