Group A has 30 students with a mean score of 50 and a standard deviation of 8. Group B has 40 students with a mean score of 55 and a standard deviation of 6. Find the combined mean and combined standard deviation.
Answer (1)
Calculate the combined mean: x ˉ co mbin e d = 30 + 40 30 × 50 + 40 × 55 = 52.86 .
Calculate the combined variance: s co mbin e d 2 = 30 + 40 − 1 ( 30 − 1 ) 8 2 + ( 40 − 1 ) 6 2 + 30 ( 50 − 52.86 ) 2 + 40 ( 55 − 52.86 ) 2 = 53.46 .
Calculate the combined standard deviation: s co mbin e d = 53.46 = 7.31 .
The combined mean is 52.86 and the combined standard deviation is 7.31 .
Explanation
Identify Given Information First, let's identify the given information:
Group A: Number of students, n A = 30 Mean score, x ˉ A = 50 Standard deviation, s A = 8
Group B: Number of students, n B = 40 Mean score, x ˉ B = 55 Standard deviation, s B = 6
Calculate the Combined Mean Next, we will calculate the combined mean using the formula: x ˉ co mbin e d = n A + n B n A x ˉ A + n B x ˉ B Substituting the given values: x ˉ co mbin e d = 30 + 40 30 × 50 + 40 × 55 = 70 1500 + 2200 = 70 3700 = 52.857142857142854 Rounding to two decimal places, the combined mean is approximately 52.86.
Calculate the Combined Variance Now, we calculate the combined variance using the formula: s co mbin e d 2 = n A + n B − 1 ( n A − 1 ) s A 2 + ( n B − 1 ) s B 2 + n A ( x ˉ A − x ˉ co mbin e d ) 2 + n B ( x ˉ B − x ˉ co mbin e d ) 2 Substituting the given values and the calculated combined mean: s co mbin e d 2 = 30 + 40 − 1 ( 30 − 1 ) 8 2 + ( 40 − 1 ) 6 2 + 30 ( 50 − 52.857142857142854 ) 2 + 40 ( 55 − 52.857142857142854 ) 2 s co mbin e d 2 = 69 29 × 64 + 39 × 36 + 30 ( − 2.857142857142854 ) 2 + 40 ( 2.142857142857146 ) 2 s co mbin e d 2 = 69 1856 + 1404 + 30 ( 8.163265306122448 ) + 40 ( 4.591836734693877 ) s co mbin e d 2 = 69 1856 + 1404 + 244.89795918367346 + 183.67346938775508 s co mbin e d 2 = 69 3688.5714285714283 = 53.457556935817806 Rounding to two decimal places, the combined variance is approximately 53.46.
Calculate the Combined Standard Deviation Finally, we calculate the combined standard deviation by taking the square root of the combined variance: s co mbin e d = s co mbin e d 2 = 53.457556935817806 = 7.3114674953676575 Rounding to two decimal places, the combined standard deviation is approximately 7.31.
State the Final Answer The combined mean is approximately 52.86, and the combined standard deviation is approximately 7.31.
Examples
Understanding combined standard deviation is useful in many real-world scenarios. For instance, consider two classes taking the same test. By calculating the combined mean and standard deviation, educators can compare the overall performance and variability between the two classes. This helps in identifying whether one class is performing significantly better than the other or if the scores are more spread out in one class compared to the other. This information can then be used to adjust teaching strategies to better meet the needs of all students.
