find the mean, median, iqr, and standard deviation given the data below. round to the nearest hundredth…

find the mean, median, iqr, and standard deviation given the data below. round to the nearest hundredth. minutes to run 5km 33.4 36.5 29.7 30.9 42.1 30.1 30.2 31.5 39 31.4 47.9 30.7 27.5 48.9 24.7 26.1 30.9 mean: median: iqr: standard deviation:

find the mean, median, iqr, and standard deviation given the data below. round to the nearest hundredth. minutes to run 5km 33.4 36.5 29.7 30.9 42.1 30.1 30.2 31.5 39 31.4 47.9 30.7 27.5 48.9 24.7 26.1 30.9 mean: median: iqr: standard deviation:

Answer

Explanation:

Step1: Calculate the mean

First, sum all the data values: $24.7+26.1+27.5+29.7+30.1+30.7+30.9+30.9+31.4+31.5+36.5+39+42.1+47.9+48.9 = 507.2$. There are $n = 15$ data - points. The mean $\bar{x}=\frac{507.2}{15}\approx33.81$.

Step2: Calculate the median

Arrange the data in ascending order: $24.7,26.1,27.5,29.7,30.1,30.7,30.9,30.9,31.4,31.5,36.5,39,42.1,47.9,48.9$. Since $n = 15$ (an odd number), the median is the $\left(\frac{n + 1}{2}\right)$-th value. $\frac{15+1}{2}=8$-th value, so the median $=30.9$.

Step3: Calculate the first and third quartiles

The lower half of the data is $24.7,26.1,27.5,29.7,30.1,30.7,30.9$. The median of the lower half (first - quartile $Q_1$) is the $\left(\frac{7 + 1}{2}\right)$-th value, which is the 4 - th value, so $Q_1=29.7$. The upper half of the data is $31.4,31.5,36.5,39,42.1,47.9,48.9$. The median of the upper half (third - quartile $Q_3$) is the $\left(\frac{7+1}{2}\right)$-th value, which is the 4 - th value of the upper half, so $Q_3 = 39$. The inter - quartile range $IQR=Q_3 - Q_1=39 - 29.7 = 9.3$.

Step4: Calculate the standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$. First, calculate $(x_i-\bar{x})^2$ for each $x_i$: $(24.7-33.81)^2=(-9.11)^2 = 82.9921$, $(26.1 - 33.81)^2=(-7.71)^2 = 59.4441$, $(27.5-33.81)^2=(-6.31)^2 = 39.8161$, $(29.7-33.81)^2=(-4.11)^2 = 16.8921$, $(30.1-33.81)^2=(-3.71)^2 = 13.7641$, $(30.7-33.81)^2=(-3.11)^2 = 9.6721$, $(30.9-33.81)^2=(-2.91)^2 = 8.4681$, $(30.9-33.81)^2=(-2.91)^2 = 8.4681$, $(31.4-33.81)^2=(-2.41)^2 = 5.8081$, $(31.5-33.81)^2=(-2.31)^2 = 5.3361$, $(36.5-33.81)^2=(2.69)^2 = 7.2361$, $(39-33.81)^2=(5.19)^2 = 26.9361$, $(42.1-33.81)^2=(8.29)^2 = 68.7241$, $(47.9-33.81)^2=(14.09)^2 = 198.5281$, $(48.9-33.81)^2=(15.09)^2 = 227.7081$. The sum $\sum_{i = 1}^{n}(x_i-\bar{x})^2=82.9921+59.4441+39.8161+16.8921+13.7641+9.6721+8.4681+8.4681+5.8081+5.3361+7.2361+26.9361+68.7241+198.5281+227.7081 = 888.8965$. Then $s=\sqrt{\frac{888.8965}{14}}\approx\sqrt{63.4926}\approx7.97$.

Answer:

Mean: $33.81$, Median: $30.9$, IQR: $9.3$, Standard Deviation: $7.97$