Question

male shoe sizes
graph: create a dotplot of the male shoe sizes.
8 8 9 9 9 9 9 9 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 14 14 14 14
mean ($\bar{x}$)=11.28 median = 11 mode = 12
q: what do you notice about all of these numbers?
shape =
standard deviation ($s_{x}$) =

Explanation:

Step1: Analyze data - count frequencies

Count the number of times each shoe - size appears in the data set. For example, size 8 appears 2 times, size 9 appears 6 times etc.

Step2: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $x_{i}$ are the individual data - points and $n$ is the total number of data - points. Here $n = 50$. $\sum_{i = 1}^{50}x_{i}=8\times2 + 9\times6+10\times6 + 11\times11+12\times12+13\times10+14\times4=564$, so $\bar{x}=\frac{564}{50}=11.28$.

Step3: Calculate the median

Arrange the data in ascending order. Since $n = 50$ (an even number), the median is the average of the 25th and 26th ordered data - points. Both the 25th and 26th values are 11, so the median is 11.

Step4: Determine the mode

The mode is the value that appears most frequently in the data set. Here, size 12 appears 12 times, more frequently than any other size, so the mode is 12.

Step5: Observe the data

The mean, median and mode are all relatively close values. This indicates that the data is somewhat symmetrically distributed.

Step6: Estimate the shape

Since the mean, median and mode are close, the shape of the distribution is approximately symmetric.

Step7: Calculate the standard deviation

The formula for the sample standard deviation $s_{x}=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$. First, calculate $(x_{i}-\bar{x})^{2}$ for each $x_{i}$, sum them up, and then divide by $n - 1=49$ and take the square - root. After calculation, $s_{x}\approx1.77$.

Answer:

Shape = Approximately symmetric
Standard Deviation $(s_{x})\approx1.77$