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Name: During a quality assurance check, the actual contents (in grams) of
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Description: During a quality assurance check, the actual contents (in grams) of six containers of protein powder were recorded as 1526, 1532, 1502, 1516 , 1534, and 1512. (a) Find the mean and the median of the contents. (b) The third value was incorrect

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

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.3: Measures Of Spread

Problem 1GP

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During a quality assurance check, the actual contents (in grams) of six containers of protein powder were recorded as 1526, 1532, 1502, 1516 , 1534, and 1512.

(a) Find the mean and the median of the contents.

(b) The third value was incorrectly measured and is actually 1517.

Find the mean and the median of the contents again.

c) Which measure of central tendency, the mean or the median, was affected more by the data entry error?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution

Step 1

From the given information

The recorded values (in grams) $x_{i} = 1526, 1532, 1502, 1516, 1534, 1512$ and n=6

Therefore the mean value is calculated as follows:

$\begin{array}{rcl} x & = & \frac{\sum_{i = 1}^{n}}{n} \\ = & \frac{1526 + 1532 + 1502 + 1516 + 1534 + 1512}{6} \\ = & 1520.33 \end{array}$

The formula for median is $M = {(\frac{n + 1}{2})}^{t h}$ observation.

Therefore,

$\begin{array}{rcl} M & = & (\frac{n + 1}{2}) \\ = & \frac{6 + 1}{2} \\ = & 3.5 \end{array}$

In this case, there are 6 observation so we have to find the average of the 3^rd and 4^th values when data is ascending order.

So, we get the median is

$\begin{array}{rcl} M & = & \frac{1516 + 1526}{2} \\ = & 1521 \end{array}$ .

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