Question

mode(s) = ______
mode(s) = ______
directions: find the mean absolute deviation, variance, and standard deviation for each data set.

1. the following data shows the number of fish caught by seven boy scouts on their camping trip:

{1, 2, 2, 4, 5, 6, 8}
mad = ______
variance: $sigma^{2}$ = ______
standard deviation: $sigma$ = ______

1. the following data shows the points scored by a football team during their first ten games of the season: {27, 32, 41, 9, 14, 20, 31, 33, 20, 13}

mad = ______
variance: $sigma^{2}$ = ______
standard deviation: $sigma$ = ______

1. the following data shows the high temperature for the past eight days:

{66, 46, 53, 50, 52, 47, 45, 49}
mad = ______
variance: $sigma^{2}$ = ______
standard deviation: $sigma$ = ______

1. the following data shows the price of six different jars of pasta sauce at the grocery store:

{$2.79, $1.99, $4.29, $2.49, $2.29, $3.49}
mad = ______
variance: $sigma^{2}$ = ______
standard deviation: $sigma$ = ______

Explanation:

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Problem 3: Data set {1, 2, 2, 4, 5, 6, 8}

Step1: Calculate the mean

$\bar{x} = \frac{1+2+2+4+5+6+8}{7} = \frac{28}{7} = 4$

Step2: Compute MAD (Mean Absolute Deviation)

Find absolute differences from mean: $|1-4|=3$, $|2-4|=2$, $|2-4|=2$, $|4-4|=0$, $|5-4|=1$, $|6-4|=2$, $|8-4|=4$
$\text{MAD} = \frac{3+2+2+0+1+2+4}{7} = \frac{14}{7} = 2$

Step3: Calculate variance $\sigma^2$

Find squared differences: $(1-4)^2=9$, $(2-4)^2=4$, $(2-4)^2=4$, $(4-4)^2=0$, $(5-4)^2=1$, $(6-4)^2=4$, $(8-4)^2=16$
$\sigma^2 = \frac{9+4+4+0+1+4+16}{7} = \frac{38}{7} \approx 5.43$

Step4: Calculate standard deviation $\sigma$

$\sigma = \sqrt{\frac{38}{7}} \approx \sqrt{5.43} \approx 2.33$

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Problem 4: Data set {27, 32, 41, 9, 14, 20, 31, 33, 20, 13}

Step1: Calculate the mean

$\bar{x} = \frac{27+32+41+9+14+20+31+33+20+13}{10} = \frac{240}{10} = 24$

Step2: Compute MAD

Find absolute differences from mean: $|27-24|=3$, $|32-24|=8$, $|41-24|=17$, $|9-24|=15$, $|14-24|=10$, $|20-24|=4$, $|31-24|=7$, $|33-24|=9$, $|20-24|=4$, $|13-24|=11$
$\text{MAD} = \frac{3+8+17+15+10+4+7+9+4+11}{10} = \frac{88}{10} = 8.8$

Step3: Calculate variance $\sigma^2$

Find squared differences: $(27-24)^2=9$, $(32-24)^2=64$, $(41-24)^2=289$, $(9-24)^2=225$, $(14-24)^2=100$, $(20-24)^2=16$, $(31-24)^2=49$, $(33-24)^2=81$, $(20-24)^2=16$, $(13-24)^2=121$
$\sigma^2 = \frac{9+64+289+225+100+16+49+81+16+121}{10} = \frac{970}{10} = 97$

Step4: Calculate standard deviation $\sigma$

$\sigma = \sqrt{97} \approx 9.85$

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Problem 5: Data set {66, 46, 53, 50, 52, 47, 45, 49}

Step1: Calculate the mean

$\bar{x} = \frac{66+46+53+50+52+47+45+49}{8} = \frac{408}{8} = 51$

Step2: Compute MAD

Find absolute differences from mean: $|66-51|=15$, $|46-51|=5$, $|53-51|=2$, $|50-51|=1$, $|52-51|=1$, $|47-51|=4$, $|45-51|=6$, $|49-51|=2$
$\text{MAD} = \frac{15+5+2+1+1+4+6+2}{8} = \frac{36}{8} = 4.5$

Step3: Calculate variance $\sigma^2$

Find squared differences: $(66-51)^2=225$, $(46-51)^2=25$, $(53-51)^2=4$, $(50-51)^2=1$, $(52-51)^2=1$, $(47-51)^2=16$, $(45-51)^2=36$, $(49-51)^2=4$
$\sigma^2 = \frac{225+25+4+1+1+16+36+4}{8} = \frac{312}{8} = 39$

Step4: Calculate standard deviation $\sigma$

$\sigma = \sqrt{39} \approx 6.24$

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Problem 6: Data set {\$2.79, \$1.99, \$4.29, \$2.49, \$2.29, \$3.49}

Step1: Calculate the mean

$\bar{x} = \frac{2.79+1.99+4.29+2.49+2.29+3.49}{6} = \frac{17.34}{6} = 2.89$

Step2: Compute MAD

Find absolute differences from mean: $|2.79-2.89|=0.1$, $|1.99-2.89|=0.9$, $|4.29-2.89|=1.4$, $|2.49-2.89|=0.4$, $|2.29-2.89|=0.6$, $|3.49-2.89|=0.6$
$\text{MAD} = \frac{0.1+0.9+1.4+0.4+0.6+0.6}{6} = \frac{4.0}{6} \approx 0.67$

Step3: Calculate variance $\sigma^2$

Find squared differences: $(2.79-2.89)^2=0.01$, $(1.99-2.89)^2=0.81$, $(4.29-2.89)^2=1.96$, $(2.49-2.89)^2=0.16$, $(2.29-2.89)^2=0.36$, $(3.49-2.89)^2=0.36$
$\sigma^2 = \frac{0.01+0.81+1.96+0.16+0.36+0.36}{6} = \frac{3.66}{6} = 0.61$

Step4: Calculate standard deviation $\sigma$

$\sigma = \sqrt{0.61} \approx 0.78$

Answer:

Problem 3:

MAD = 2, Variance $\sigma^2 = \frac{38}{7} \approx 5.43$, Standard Deviation $\sigma \approx 2.33$

Problem 4:

MAD = 8.8, Variance $\sigma^2 = 97$, Standard Deviation $\sigma \approx 9.85$

Problem 5:

MAD = 4.5, Variance $\sigma^2 = 39$, Standard Deviation $\sigma \approx 6.24$

Problem 6:

MAD $\approx 0.67$, Variance $\sigma^2 = 0.61$, Standard Deviation $\sigma \approx 0.78$