Understanding the Mean and Standard Deviation

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Question 1 of 2

1. Question
The height of a group of $5$ dogs and a group of $5$ cats are collected in centimetres. Compare the mean for dogs $bar x$ _D and the mean for cats $bar x$ _C.
- $bar x$ _D $=$ (40) $cm$
  
  $bar x$ _C $=$ (40) $cm$
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Formula for Mean

$bar x=(\text(sum of data))/(\text(number of data))$

$bar x$ _D $=$ dogs mean

$bar x$ _C $=$ cats mean

Use the mean formula to find the average height of the dogs.

$bar x$ _D $=$ $(\text(sum of data))/(\text(number of data))$

$=$ $(20+30+40+50+60)/5$ Substitute values

$=$ $200/5$

$=$ $40$ $cm$ Simplify

Use the mean formula to find the average height of the cats.

$bar x$ _C $=$ $(\text(sum of data))/(\text(number of data))$

$=$ $(38+39+40+41+42)/5$ Substitute values

$=$ $200/5$

$=$ $40$ $cm$ Simplify

The mean of both groups is $40$ cm even though their heights are not the same.

This only means that both groups have their data centralized or averaged at $40$ cm.

$bar x$ _D $=40$ $cm$

$bar x$ _C $=40$ $cm$

Question 2 of 2

The height of a group of

$5$ dogs and a group of

$5$ cats are collected in centimetres. Find and compare the standard deviation of both groups.

The mean of both groups is given:

$barX=40$

$S_d=$ (14.1) $cm$

$S_c=$ (1.4) $cm$

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Formula for Standard Deviation

$S=\sqrt{\frac{\sum(\color{#9a00c7}{X}-\color{#00880a}{\bar{X)}}^2}{\color{#004ec4}{n}}}$

To solve for the standard deviation for the height of the dogs, label the given values needed for the formula.

$X \text(values) \text((heights))=20,30,40,50,60$

$\text(Mean) barX=40$

$\text(Number of dogs): n=5$

Subsitute into the standard deviation formula

$S$	$=$	$\sqrt{\frac{\sum(\color{#9a00c7}{X}-\color{#00880a}{\bar{X)}}^2}{\color{#004ec4}{n}}}$

	$=$	$\sqrt{\frac{(\color{#9a00c7}{20}-\color{#00880a}{40})^2+(\color{#9a00c7}{30}-\color{#00880a}{40})^2+(\color{#9a00c7}{40}-\color{#00880a}{40})^2+(\color{#9a00c7}{50}-\color{#00880a}{40})^2+(\color{#9a00c7}{60}-\color{#00880a}{40})^2}{\color{#004ec4}5}}$	Substitute values

	$=$	$sqrt(((-20)^2+(-10)^2+(0)^2+(10)^2+(20)^2)/5)$

	$=$	$sqrt((400+100+0+100+400)/5)$

	$=$	$sqrt(1000/5)$

	$=$	$sqrt(200)$
$S_d$	$=$	$14.1$

To solve for the standard deviation for the height of the cats, label the given values needed for the formula.

$X \text(values) \text((heights))=38,39,40,41,42$

$\text(Mean) barX=40$

$\text(Number of cats): n=5$

Subsitute into the standard deviation formula

$S$	$=$	$\sqrt{\frac{\sum(\color{#9a00c7}{X}-\color{#00880a}{\bar{X)}}^2}{\color{#004ec4}{n}}}$

	$=$	$\sqrt{\frac{(\color{#9a00c7}{38}-\color{#00880a}{40})^2+(\color{#9a00c7}{39}-\color{#00880a}{40})^2+(\color{#9a00c7}{40}-\color{#00880a}{40})^2+(\color{#9a00c7}{41}-\color{#00880a}{40})^2+(\color{#9a00c7}{42}-\color{#00880a}{40})^2)}{\color{#004ec4}5}}$	Substitute values

	$=$	$sqrt(((-2)^2+(-1)^2+(0)^2+(1)^2+(2)^2)/5)$

	$=$	$sqrt((4+1+0+1+4)/5)$

	$=$	$sqrt(10/5)$	Simplify

	$=$	$sqrt2$
$S_c$	$=$	$1.4$

$S_d$ is greater than

$S_c$ .

This means that the height of the dogs is more dispersed than the height of the cats.

$S_d=14.1cm$

$S_c=1.4cm$

Understanding the Mean and Standard Deviation

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Quiz summary

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1. Question

Hint

Fantastic!

Formula for Mean

2. Question

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Formula for Standard Deviation

Quizzes

$bar x$ _D	$=$	$(\text(sum of data))/(\text(number of data))$

	$=$	$(20+30+40+50+60)/5$	Substitute values

	$=$	$200/5$

	$=$	$40$ $cm$	Simplify

$bar x$ _C	$=$	$(\text(sum of data))/(\text(number of data))$

	$=$	$(38+39+40+41+42)/5$	Substitute values

	$=$	$200/5$

	$=$	$40$ $cm$	Simplify