Suppose we have the following dataset representing the ages of 10 students in a class: 21, 22, 20, 23, 19, 21, 25, 24, 20, 22. Calculate the mean, median, mode, variance, and standard deviation of the ages.
Solution:
Mean:
To find the mean (average) age_Mean=?agesnumber of students=21+22+20+23+19+21+25+24+20+2210=21710=21.7text{Mean} = frac{sum text{ages}}{text{number of students}} = frac{21 + 22 + 20 + 23 + 19 + 21 + 25 + 24 + 20 + 22}{10} = frac{217}{10} = 21.7Mean=number of students?ages=1021+22+20+23+19+21+25+24+20+22=10217=21.7
Median:
To find the median (middle value when arranged in order):
First, arrange the ages in ascending order: 19, 20, 20, 21, 21, 22, 22, 23, 24, 25.
The median is the average of the 5th and 6th values_Median=21+222=21.5text{Median} = frac{21 + 22}{2} = 21.5Median=221+22=21.5
Mode:
The mode is the most frequent age:
In this dataset, 21 and 22 both occur twice, so the modes are 21 and 22.
Variance:
To find the variance (average of the squared differences from the mean):
First, calculate the squared differences from the mean:Squared differences: (21?21.7)2,
(22?21.7)2,Â…,(22?21.7)2text{Squared differences: } (21-21.7)^2, (22-21.7)^2, ldots, (22-21.7)^2Squared differences: (21?21.7)2,(22?21.7)2,Â…,(22?21.7)2Sum these squared differences:Sum of squared differences=38.1text{Sum of squared differences} = 38.1Sum of squared differences=38.1
Variance=Sum of squared differencesnumber of students?1=38.19=4.2333text{Variance} = frac{text{Sum of squared differences}}{text{number of students} – 1} = frac{38.1}{9} = 4.2333Variance=number of students?1Sum of squared differences=938.1=4.2333
Standard Deviation:
Finally, calculate the standard deviation (square root of varianceStandard Deviation=4.2333?2.06text{Standard Deviation} = sqrt{4.2333} approx 2.06Standard Deviation=4.2333?2.06
Conclusion:
Mean age: 21.7 years
Median age: 21.5 years
Mode(s): 21 and 22 years
Variance: 4.2333 (square of the standard deviation)
Standard Deviation: Approximately 2.06 years
These statistics provide a summary of the ages of the students in the class, showing the central tendency (mean, median, mode) and the variability (variance, standard deviation) of the dataset.