Question

(1 point) consider the following data set. find the mean and standard deviation. data set: 80, 26, 78, 66, 48, 84, 58 mean: standard deviation: if the data value 21 was added to the set, would the standard deviation become larger (l) or smaller (s). you do not need to calculate this second standard deviation to answer this question. l or s: note: you can earn partial credit on this problem you are in the reduced scoring period: all additional work done counts 85% of the original. preview my answers submit answers you have attempted this problem 0 times. you have 5 attempts left before new version will be requested. you have unlimited attempts remaining.

Explanation:

Step1: Calculate the mean

The mean ($\bar{x}$) of a data set is the sum of all values divided by the number of values. The data set is \(80, 26, 78, 66, 48, 84, 58\), and there are \(n = 7\) values.
First, find the sum: \(80 + 26 + 78 + 66 + 48 + 84 + 58\)
\(80+26 = 106\), \(106+78 = 184\), \(184+66 = 250\), \(250+48 = 298\), \(298+84 = 382\), \(382+58 = 440\)
Then, the mean is \(\bar{x}=\frac{440}{7}\approx62.86\) (rounded to two decimal places)

Step2: Calculate the standard deviation

The formula for the sample standard deviation (since this is a data set, we'll assume sample) is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)
First, find each \((x_{i}-\bar{x})^{2}\):

• For \(x_1 = 80\): \((80 - 62.86)^{2}=(17.14)^{2}\approx293.78\)
• For \(x_2 = 26\): \((26 - 62.86)^{2}=(- 36.86)^{2}\approx1358.66\)
• For \(x_3 = 78\): \((78 - 62.86)^{2}=(15.14)^{2}\approx229.22\)
• For \(x_4 = 66\): \((66 - 62.86)^{2}=(3.14)^{2}\approx9.86\)
• For \(x_5 = 48\): \((48 - 62.86)^{2}=(-14.86)^{2}\approx220.82\)
• For \(x_6 = 84\): \((84 - 62.86)^{2}=(21.14)^{2}\approx446.80\)
• For \(x_7 = 58\): \((58 - 62.86)^{2}=(-4.86)^{2}\approx23.62\)

Now, sum these squared differences: \(293.78+1358.66 + 229.22+9.86+220.82+446.80+23.62\)
\(293.78+1358.66 = 1652.44\), \(1652.44+229.22 = 1881.66\), \(1881.66+9.86 = 1891.52\), \(1891.52+220.82 = 2112.34\), \(2112.34+446.80 = 2559.14\), \(2559.14+23.62 = 2582.76\)

Then, divide by \(n - 1=6\): \(\frac{2582.76}{6}\approx430.46\)
Take the square root: \(s=\sqrt{430.46}\approx20.75\) (rounded to two decimal places)

Step3: Determine the effect on standard deviation

Standard deviation measures the spread of data. Adding a value (21) that is far from the mean (around 62.86) will increase the spread, so the standard deviation will become larger (L).

Answer:

Mean: \(\approx62.86\)
Standard deviation: \(\approx20.75\)
L or S: L