Question

find each item listed below for the data provided. test scores: 50, 69, 70, 75, 75, 90, 85, 80, 70, 60, 100, 95, 90, 80, 85, 50 (all answers should be rounded to the nearest tenth) mean: median: range: iqr: standard deviation:

Explanation:

Step1: Arrange data in ascending order

50, 50, 60, 69, 70, 70, 75, 75, 80, 80, 85, 85, 90, 90, 95, 100

Step2: Calculate the mean

Sum of data = \(50 + 50+60 + 69+70+70+75+75+80+80+85+85+90+90+95+100 = 1184\)
Number of data points \(n = 16\)
Mean \(\bar{x}=\frac{1184}{16}=74\)

Step3: Calculate the median

Since \(n = 16\) (even), median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th ordered - data values.
\(\frac{n}{2}=8\) and \(\frac{n}{2}+1 = 9\)
The 8th value is 75 and the 9th value is 80.
Median \(M=\frac{75 + 80}{2}=77.5\)

Step4: Calculate the range

Range = Maximum value - Minimum value
Range \(=100 - 50=50\)

Step5: Calculate the quartiles

The lower half of the data is 50, 50, 60, 69, 70, 70, 75, 75
The median of the lower half (\(Q_1\)) is the average of the 4th and 5th values.
\(Q_1=\frac{69+70}{2}=69.5\)
The upper half of the data is 80, 80, 85, 85, 90, 90, 95, 100
The median of the upper half (\(Q_3\)) is the average of the 4th and 5th values.
\(Q_3=\frac{85 + 90}{2}=87.5\)
Inter - quartile range \(IQR=Q_3 - Q_1=87.5-69.5 = 18\)

Step6: Calculate the standard deviation

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=(50 - 74)^2+(50 - 74)^2+(60 - 74)^2+(69 - 74)^2+(70 - 74)^2+(70 - 74)^2+(75 - 74)^2+(75 - 74)^2+(80 - 74)^2+(80 - 74)^2+(85 - 74)^2+(85 - 74)^2+(90 - 74)^2+(90 - 74)^2+(95 - 74)^2+(100 - 74)^2\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=(-24)^2+(-24)^2+(-14)^2+(-5)^2+(-4)^2+(-4)^2+(1)^2+(1)^2+(6)^2+(6)^2+(11)^2+(11)^2+(16)^2+(16)^2+(21)^2+(26)^2\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2 = 576+576+196 + 25+16+16+1+1+36+36+121+121+256+256+441+676\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=3696\)
\(s=\sqrt{\frac{3696}{15}}\approx15.7\)

Answer:

Mean: 74
Median: 77.5
Range: 50
IQR: 18
Standard Deviation: 15.7