1 the shoe size for all the pairs of shoes in a person’s closet are recorded.
7 7 7 7 7 7 7 7 7 7
a. what is the mean?
b. what is the standard deviation?
2 here is a data set:
1 2 3 3 4 4 4 4 5 5 6 7
a. what happens to the mean and standard deviation of the data set when the 7 is changed to a 70?
b. for the data set with the value of 70, why would the median be a better choice for the measure of center than the mean?
3 which of these best estimates the standard deviation of points in a card game?
a 5 points
b 20 points
c 50 points
d 500 points
(chart: dot plot with points at 490, 495, 500, 505, 510, labeled points in a card game)

Question

Accepted Answer

### Problem 1
#### Part a: Mean Calculation
# Explanation:
## Step1: Recall the mean formula
The mean $\bar{x}$ of a data set is calculated as $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $x_{i}$ are the data points and $n$ is the number of data points.
Here, all the data points are 7, and there are $n = 10$ data points. So $\sum_{i=1}^{10}x_{i}=7\times10 = 70$.
## Step2: Calculate the mean
Using the formula, $\bar{x}=\frac{70}{10}=7$.
# Answer: 7

#### Part b: Standard Deviation Calculation
# Explanation:
## Step1: Recall the standard deviation formula for a population
The population standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$. Since all $x_{i}=7$ and $\bar{x}=7$, then $x_{i}-\bar{x}=0$ for all $i$.
## Step2: Calculate the sum of squared deviations
$\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=\sum_{i = 1}^{10}(7 - 7)^{2}=\sum_{i = 1}^{10}0=0$.
## Step3: Calculate the standard deviation
$\sigma=\sqrt{\frac{0}{10}} = 0$.
# Answer: 0

### Problem 2
#### Part a: Effect on Mean and Standard Deviation
# Explanation:
## Step1: Effect on Mean
The mean is $\bar{x}=\frac{\sum x_{i}}{n}$. Originally, the sum of the data set (excluding 7 for a moment) is $1 + 2+3 + 3+4 + 4+4 + 4+5 + 5+6=1 + 2+6 + 8+16 + 10=43$. With 7, the sum is $43 + 7=50$ and $n = 12$. After changing 7 to 70, the new sum is $43+70 = 113$. The new mean is $\frac{113}{12}\approx9.42$, which is an increase from the original mean $\frac{50}{12}\approx4.17$.
## Step2: Effect on Standard Deviation
Standard deviation measures the spread of data. The original data point 7 is close to the other data points. When we change it to 70, which is far from the other data points, the spread of the data increases. So the standard deviation will increase.
# Answer: The mean increases, and the standard deviation increases.

#### Part b: Median vs Mean as Measure of Center
# Explanation:
The median is the middle value of a data set (or the average of the two middle values) when the data is ordered. The mean is affected by extreme values (outliers). In this data set, 70 is an outlier (a value much larger than the other data points). The mean is pulled up by the outlier 70, but the median is not affected by extreme values. So the median gives a more representative measure of the center of the data set as it is resistant to outliers, while the mean is not.
# Answer: The median is better because it is resistant to outliers (the 70 is an outlier that skews the mean, but the median remains unaffected by this extreme value and better represents the center of the majority of the data).

### Problem 3
# Explanation:
Standard deviation measures the typical distance of data points from the mean. Looking at the dot - plot, the data points are around 490, 495, 500, 505, 510. The spread between the data points is relatively small. 50 points and 500 points are too large, and 20 points is also larger than the typical spread. The data points are within a range of about 20 (from 490 to 510), but the typical distance from the mean (around 500) is about 5 points. So 5 points is the best estimate.
# Answer: A. 5 points

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QUESTION IMAGE

Question

Explanation:

Problem 1

Part a: Mean Calculation

Step1: Recall the mean formula

Step2: Calculate the mean

Step1: Recall the standard deviation formula for a population

Step2: Calculate the sum of squared deviations

Step3: Calculate the standard deviation

Step1: Effect on Mean

Step2: Effect on Standard Deviation

Answer:

Part b: Standard Deviation Calculation