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QuestionCalculate the mean, median, standard deviation, and range of these weights: 2.1, 4.2, 3.4, 0.8, 4.2, 3.1, 2, 3, 2.6, 1.4. Round to three decimal places.

Studdy Solution

STEP 1

Assumptions1. The weights of the wireless routers are given in the problem. . We are asked to find the mean, median, standard deviation, and range of the weights.
3. We will round our answers to three decimal places where necessary.

STEP 2

First, let's find the mean. The mean is the sum of all the weights divided by the number of weights.
Mean=Sum of all weightsNumber of weightsMean = \frac{Sum\, of\, all\, weights}{Number\, of\, weights}

STEP 3

Now, plug in the given weights to calculate the sum.
Sum of all weights=2.1+.2+3.+0.8+.2+3.1+2+3+2.6+1.Sum\, of\, all\, weights =2.1 +.2 +3. +0.8 +.2 +3.1 +2 +3 +2.6 +1.

STEP 4

Calculate the sum of all weights.
Sum of all weights=26.8Sum\, of\, all\, weights =26.8

STEP 5

Now, divide the sum of all weights by the number of weights to calculate the mean.
Mean=26.810Mean = \frac{26.8}{10}

STEP 6

Calculate the mean.
Mean=2.68Mean =2.68

STEP 7

Next, let's find the median. The median is the middle number when the weights are arranged in ascending order. If there is an even number of weights, the median is the average of the two middle numbers.
First, arrange the weights in ascending order.
0.,1.4,2,2.1,2.6,3,3.1,3.4,4.2,4.20.,1.4,2,2.1,2.6,3,3.1,3.4,4.2,4.2

STEP 8

Since there are10 weights (an even number), the median is the average of the5th and6th weights.
Median=5th weight+6th weight2Median = \frac{5th\, weight +6th\, weight}{2}

STEP 9

Plug in the values for the5th and6th weights to calculate the median.
Median=2.6+32Median = \frac{2.6 +3}{2}

STEP 10

Calculate the median.
Median=2.8Median =2.8

STEP 11

Next, let's find the standard deviation. The standard deviation measures the amount of variation or dispersion of a set of values.First, we need to find the variance. The variance is the average of the squared differences from the mean.
Variance = \frac{Sum\, of\, (each\, weight - Mean)^}{Number\, of\, weights}

STEP 12

Now, plug in the given weights and the mean to calculate the sum of the squared differences.
Sum of (each weight−Mean)2=(2.−2.68)2+(4.2−2.68)2+(.4−2.68)2+(0.8−2.68)2+(4.2−2.68)2+(.−2.68)2+(2−2.68)2+(−2.68)2+(2.6−2.68)2+(.4−2.68)2Sum\, of\, (each\, weight - Mean)^2 = (2. -2.68)^2 + (4.2 -2.68)^2 + (.4 -2.68)^2 + (0.8 -2.68)^2 + (4.2 -2.68)^2 + (. -2.68)^2 + (2 -2.68)^2 + ( -2.68)^2 + (2.6 -2.68)^2 + (.4 -2.68)^2

STEP 13

Calculate the sum of the squared differences.
Sum of (each weight−Mean)2=0.336+2.310+0.518+3.5328+2.310+0.1936+0.462+0.102+0.006+.638Sum\, of\, (each\, weight - Mean)^2 =0.336 +2.310 +0.518 +3.5328 +2.310 +0.1936 +0.462 +0.102 +0.006 +.638

STEP 14

Now, divide the sum of the squared differences by the number of weights to calculate the variance.
Variance=11.411610Variance = \frac{11.4116}{10}

STEP 15

Calculate the variance.
Variance=.141Variance =.141

STEP 16

The standard deviation is the square root of the variance.
Standard Deviation=VarianceStandard\, Deviation = \sqrt{Variance}

STEP 17

Plug in the value for the variance to calculate the standard deviation.
Standard Deviation=.14116Standard\, Deviation = \sqrt{.14116}

STEP 18

Calculate the standard deviation.
Standard Deviation=.068Standard\, Deviation =.068

STEP 19

Finally, let's find the range. The range is the difference between the highest and lowest weights.
Range=Highest weight−Lowest weightRange = Highest\, weight - Lowest\, weight

STEP 20

Plug in the values for the highest and lowest weights to calculate the range.
Range=4.−0.8Range =4. -0.8

STEP 21

Calculate the range.
Range=3.4Range =3.4So, the mean is.68, the median is.8, the standard deviation is1.068, and the range is3.4.

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