Question

Two sample sizes of 100 and 150 respectively have means 50 and 60 and standard deviations 5 and 6.​

Answer 1

Explanation:

Two sample of size 100 and 150 respectively have means 50 and 60 deviation of the combined sample of size 250. Hence, the Combined Mean =56 and the Combined Standard Deviation=7.46

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Answer 2

The complete question: Two sample of sizes 100 and 150 respectively have means 50 and 60 and standard deviation 5 and 6.Find the combined mean and combined standard deviation.

Given:

Two sample sizes of 100 and 150 respectively.

Mean of the two samples are 50 and 60 respectively.

Standard deviations of the two samples are 5 and 6 respectively.

To Find:

Combined mean and Combined standard deviation.

Solution:

Let m be the number of elements in the first sample.

Let n be the number of elements in the second sample.

Let x1 be the mean of the first sample

Let x2 be the mean of the second sample.

Let Xc be the combined mean.

• Simply put, a combined mean is a weighted mean with each group's size serving as the weight.
• A combined mean is the average value obtained by calculating the means of two or more independent groups and then combining the results.

Now, The formula for calculating the combined mean is as follows:

$Xc = \frac{mx1 + nx2}{m+n}$

Xc = 50×100 + 60×150/ 100 + 150

Xc = 5000 + 9000/ 250

Xc = 14000/250

Xc = 56

Let standard deviations of first sample and second sample be s1 and s2, respectively. Xc is the combined mean and Let Sc is the combined standard deviation of the two samples.

• The square root of the combined variance yields the combined standard deviation.

Now, The formula for calculating the combined standard deviation is as follows:

Sc = $\sqrt[]{\frac{m[s1 + (Xc - x1)^2] + m[s2 + (Xc - x2)^2]}{ m+n} }$

Sc = √ 100×[5 + (56-50)²] + 150×[6 + (56 - 60)²] / 250

Sc = √ (100 × 41 + 150 × 22) / 250

Sc = √4100 + 3300 / 250

Sc = 5.44

Hence, the combined mean and standard deviation are 56 and 5.44, respectively.

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