Question

Listed below are the amounts​ (dollars) it costs for marriage proposal packages at different baseball stadiums. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are there any​ outliers, and are they likely to have much of an effect on the measures of​ variation? 40 45 50 65 65 80 85 105 190 212 250 375 500 1750 2500

Answer 1

Answer: Range = 2460 dollars; Variance (s²) = 516414.6; Standard Deviation (s) = 718.62

Step-by-step explanation: Range is the difference between the lowest and highest values of the set. For this data set:

Range = 2500 - 40

Range = 2460

Variance is the average of the squared differences from the mean, so first you calculate the mean of the data:

μ = ∑x / N

μ = $\frac{40+45+50+65+65+80+85+105+190+212+250+375+500+1750+2500}{15}$

μ = 420.8 dollars

With the mean, calculate the variance:

s² = [∑(x - μ)²] / N - 1

s² = $\frac{(40 - 420.8)^{2} + (45 - 420.8)^{2} + ... + (2500 - 420.8)^{2} }{15 - 1}$

s² = 516414.6 dollars

Note: To calculate variance you have to subtract each value from the data with the mean found, square the difference and then add all the squares.

Standard Deviation is how spread the numbers are. It's calculated as the square root of variance:

s = $\sqrt{s^{2} }$

s = $\sqrt{516414.6}$

s = 718.62 dollars

Outliers are values that are too high or too low from the other values. In this data set the packages which costs 1750 and 2500 are too high compared to the others. So, those are the outliers of this data. They affect the mean of the data, and mean is important in variance and standard devation. Therefore, outliers have a great effect on variance.

Answer 2

Answer:

Range = 2460 dollars, Variance = 516414.6 $dollars^2$, Standard deviation = 718.6199 dollars . There are two outliers and they are likely to have much of an effect on the measures of variation.

Step-by-step explanation:

The smallest value in the sample data is min = 50 dollars and the largest value is max = 2500 dollars, therefore, the range is Range = max - min = 2500 - 40 = 2460 dollars. On the other hand, the formula to compute the sample variance is $S^2 = \frac{1}{n-1}\sum_{i=1}^{n} (x_{i}-\bar{x})^2$ where $\bar{x}$ is the sample mean, n is the sample size and the $x_{i}$ are the sample values. In this case the sample variance is $s^2$ = 516414.6 $dollars^2$, the sample standard deviation is defined as the squared root of the sample variance, so, the sample standard deviation is s = 718.6199 dollars. There are two outliers because 1750 dollars and 2500 dollars are very different compared to the other values, these two numbers are very large and they are likely to have much of an effect on the measures of variation because these measures are sensible to outliers, they are no robust measures.