Questions: Which list appears to have the greatest standard deviation? a 98, 98, 100, 97, 92, 104 b 100, 100, 100, 100, 100, 100 c 99, 107, 83, 92, 86, 90 d 142, 95, 164, 125, 169, 128

Which list appears to have the greatest standard deviation? a 98, 98, 100, 97, 92, 104 b 100, 100, 100, 100, 100, 100 c 99, 107, 83, 92, 86, 90 d 142, 95, 164, 125, 169, 128

Transcript text: Which list appears to have the greatest standard deviation? a $98,98,100,97,92,104$ b $100,100,100,100,100,100$ c $99,107,83,92,86,90$ d $142,95,164,125,169,128$

Solution

Solution Steps

Step 1: Understand the concept of standard deviation

Standard deviation measures the amount of variation or dispersion in a set of values. A higher standard deviation indicates that the values are more spread out from the mean, while a lower standard deviation indicates that the values are closer to the mean.

Step 2: Calculate the mean for each list

For each list, calculate the mean ($\mu$) using the formula: \[ \mu = \frac{\sum x_i}{n} \] where $x_i$ are the values in the list and $n$ is the number of values.

List a: $98, 98, 100, 97, 92, 104$ \[ \mu_a = \frac{98 + 98 + 100 + 97 + 92 + 104}{6} = \frac{589}{6} = 98.1667 \]
List b: $100, 100, 100, 100, 100, 100$ \[ \mu_b = \frac{100 + 100 + 100 + 100 + 100 + 100}{6} = \frac{600}{6} = 100 \]
List c: $99, 107, 83, 92, 86, 90$ \[ \mu_c = \frac{99 + 107 + 83 + 92 + 86 + 90}{6} = \frac{557}{6} = 92.8333 \]
List d: $142, 95, 164, 125, 169, 128$ \[ \mu_d = \frac{142 + 95 + 164 + 125 + 169 + 128}{6} = \frac{823}{6} = 137.1667 \]

Step 3: Calculate the standard deviation for each list

The standard deviation ($\sigma$) is calculated using the formula: \[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}} \]

List a: \[ \sigma_a = \sqrt{\frac{(98-98.1667)^2 + (98-98.1667)^2 + (100-98.1667)^2 + (97-98.1667)^2 + (92-98.1667)^2 + (104-98.1667)^2}{6}} \] \[ \sigma_a = \sqrt{\frac{0.0278 + 0.0278 + 3.3611 + 1.3611 + 38.0278 + 34.0278}{6}} = \sqrt{\frac{76.8334}{6}} = \sqrt{12.8056} = 3.5785 \]
List b: \[ \sigma_b = \sqrt{\frac{(100-100)^2 + (100-100)^2 + (100-100)^2 + (100-100)^2 + (100-100)^2 + (100-100)^2}{6}} = \sqrt{\frac{0}{6}} = 0 \]
List c: \[ \sigma_c = \sqrt{\frac{(99-92.8333)^2 + (107-92.8333)^2 + (83-92.8333)^2 + (92-92.8333)^2 + (86-92.8333)^2 + (90-92.8333)^2}{6}} \] \[ \sigma_c = \sqrt{\frac{38.0278 + 201.3611 + 96.3611 + 0.6944 + 46.6944 + 8.0278}{6}} = \sqrt{\frac{391.1666}{6}} = \sqrt{65.1944} = 8.0743 \]
List d: \[ \sigma_d = \sqrt{\frac{(142-137.1667)^2 + (95-137.1667)^2 + (164-137.1667)^2 + (125-137.1667)^2 + (169-137.1667)^2 + (128-137.1667)^2}{6}} \] \[ \sigma_d = \sqrt{\frac{23.3611 + 1778.0278 + 720.0278 + 148.0278 + 1014.0278 + 84.0278}{6}} = \sqrt{\frac{3767.5001}{6}} = \sqrt{627.9167} = 25.0583 \]

Step 4: Compare the standard deviations