Question

1.46/47 means, medians, standard deviations, and iqrs: answer the following about each dataset. (round to two decimal places where appropriate)
dataset i: 0 10 50 60 100
a) the median of dataset i is:
b) the iqr of dataset i is from to
c) the mean of dataset i is:
d) the standard deviation of dataset i is:
dataset ii: 0 100 500 600 1000
e) the median of dataset ii is:
f) the iqr of dataset ii is from to
g) the mean of dataset ii is:
h) the standard deviation of dataset ii is:
question help: video

Explanation:

Step1: Find median of Dataset I

Dataset I has 5 data - points. For odd - numbered data, median is the middle value. So, median = 50.

Step2: Find quartiles of Dataset I

First, arrange data in ascending order: 0, 10, 50, 60, 100. The first quartile $Q_1$ is the median of the lower half (0, 10), so $Q_1=\frac{0 + 10}{2}=5$. The third quartile $Q_3$ is the median of the upper half (60, 100), so $Q_3=\frac{60+100}{2}=80$. The IQR = $Q_3 - Q_1=80 - 5 = 75$, from 5 to 80.

Step3: Calculate mean of Dataset I

Mean $\bar{x}=\frac{0 + 10+50+60+100}{5}=\frac{220}{5}=44$.

Step4: Calculate standard deviation of Dataset I

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$.
$(0 - 44)^2=1936$, $(10 - 44)^2 = 1156$, $(50 - 44)^2=36$, $(60 - 44)^2 = 256$, $(100 - 44)^2=3136$.
$\sum_{i = 1}^{5}(x_i - 44)^2=1936+1156+36+256+3136 = 6520$.
$s=\sqrt{\frac{6520}{4}}\approx40.37$.

Step5: Find median of Dataset II

Dataset II has 5 data - points. Median is the middle value, so median = 500.

Step6: Find quartiles of Dataset II

Arrange data in ascending order: 0, 100, 500, 600, 1000. $Q_1=\frac{0 + 100}{2}=50$, $Q_3=\frac{600+1000}{2}=800$. IQR = $Q_3 - Q_1=800 - 50 = 750$, from 50 to 800.

Step7: Calculate mean of Dataset II

Mean $\bar{x}=\frac{0+100 + 500+600+1000}{5}=\frac{2200}{5}=440$.

Step8: Calculate standard deviation of Dataset II

$(0 - 440)^2=193600$, $(100 - 440)^2 = 115600$, $(500 - 440)^2=3600$, $(600 - 440)^2 = 25600$, $(1000 - 440)^2=313600$.
$\sum_{i = 1}^{5}(x_i - 440)^2=193600+115600+3600+25600+313600 = 652000$.
$s=\sqrt{\frac{652000}{4}}\approx403.73$.

Answer:

a) 50
b) 5, 80
c) 44
d) 40.37
e) 500
f) 50, 800
g) 440
h) 403.73