find each item listed below for the data provided. test scores: 50, 69, 70, 75, 75, 90, 85, 80, 70, 60, 100…

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