Two data set of sizes 9 and 6 have standard deviation 3 and 4 respectively and arithmetic means 3 respectively. The standard deviation of combined data set of size 15 is

The correct answer is √177/15.

Alternate Method

The variance of the group square it's standard deviation.

For the first group (size = 9, standard deviation = 3)

Variance = 3² = 9

For the second group (size = 6, standard deviation = 4)

Variance = 4² = 16

Combined variance = (9 × 9 + 6 × 16)/15 = 11.8

Standard deviation = √(11.8)

 ⇒ √(177/15) is the correct answer.  

Key Points 

• It's important to note that the standard deviation is a measure of the dispersion of a set of data.
• The larger the standard deviation, the more spread out the data is.
• In this case, the standard deviation of the combined data set of size 15 is calculated by taking into account the standard deviations and sample sizes of the two original data sets.
• By combining the two data sets, the standard deviation of the combined data set takes into account the dispersion of both original data sets and provides a more accurate representation of the spread of the combined data.

 Additional Information

• To find the standard deviation of the combined data set, we can use the formula for the weighted average of the standard deviations of two data sets:
• σ = √[(n1-1)σ1^2 + (n2-1)σ2^2] / (n1 + n2 - 2)
• where n1 and n2 are the sizes of the two data sets and σ1 and σ2 are their standard deviations.
• Plugging in the given values, we get:
• σ = √[(9-1)3^2 + (6-1)4^2] / (9 + 6 - 2) = √[72 + 72] / 11 = √144 / 11 = √177 / 15
• So, the standard deviation of the combined data set of size 15 is √177/15.
• Additionally, it's also important to note that this calculation assumes that the two data sets are independent and not related.
• If the two data sets are related in some way, the calculation would be different and may require more information to be accurately calculated.