Определить среднее и стандартное отклонения для серии из четырех определений, которые дали следующие результаты: $18,50$ ; $18,68$ ; $18,43$ ; $18,70$ .

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Grade 8

Identify equivalent linear expressions: word problems

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Q. Определить среднее и стандартное отклонения для серии из четырех определений, которые дали следующие результаты:

18,50

;

18,68

;

18,43

;

18,70

Calculate Mean: Calculate the mean (average) of the four measurements. $\newline$ To find the mean, add up all the measurements and divide by the number of measurements. $\newline$ Mean = $(18.50 + 18.68 + 18.43 + 18.70) / 4$ $\newline$ = $74.31 / 4$ $\newline$ = $18.5775$
Calculate Deviations: Calculate the deviations of each measurement from the mean. To find the deviations, subtract the mean from each measurement. Deviation for $18.50 = 18.50 - 18.5775 = -0.0775$ Deviation for $18.68 = 18.68 - 18.5775 = 0.1025$ Deviation for $18.43 = 18.43 - 18.5775 = -0.1475$ Deviation for $18.70 = 18.70 - 18.5775 = 0.1225$
Square Deviations: Square each of the deviations to prepare for calculating the variance. $\newline$ $(\text{Deviation for } 18.50)^2 = (-0.0775)^2 = 0.00600625$ $\newline$ $(\text{Deviation for } 18.68)^2 = (0.1025)^2 = 0.01050625$ $\newline$ $(\text{Deviation for } 18.43)^2 = (-0.1475)^2 = 0.02175625$ $\newline$ $(\text{Deviation for } 18.70)^2 = (0.1225)^2 = 0.01500625$
Sum of Squared Deviations: Calculate the sum of the squared deviations. $\newline$ Sum of squared deviations = $0.00600625 + 0.01050625 + 0.02175625 + 0.01500625$ $\newline$ = $0.053275$
Calculate Variance: Calculate the variance. $\newline$ Since we have a small sample size $n=4$ , we will use the sample variance formula, which divides by $(n-1)$ instead of $n$ . $\newline$ Variance = Sum of squared deviations / $(n - 1)$ $\newline$ = $0.053275 / (4 - 1)$ $\newline$ = $0.053275 / 3$ $\newline$ = $0.0177583333$
Calculate Standard Deviation: Calculate the standard deviation. The standard deviation is the square root of the variance. Standard Deviation = $\sqrt{\text{Variance}} = \sqrt{0.0177583333} = 0.133256$