Question

The standard deviation for the scores \(1,2,3,4,5,6\) and 7 is 2 then the standard deviation of \(13,24,35,46,57,68\) and 79 is (a) 2 (b) 22 (c) 11 (d) 23

Short answer

The standard deviation of the second dataset is 22. Hence, the correct answer is (b) 22.

Step-by-step solution

Find the standard deviation of the first set

To find the standard deviation of the first set, we will use the formula: \[ \sigma = \sqrt{\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n}} \] The mean, \(\bar{x}\), of the first set of values is the sum of the values divided by the number of values: \[ \bar{x} = \frac{1+2+3+4+5+6+7}{7} = \frac{28}{7} = 4 \] We are given that the standard deviation of the first set is 2.

Examine the relationship between the two set of values

Notice that the second set of values is obtained by multiplying each value in the first set by 11: \[ 11*1 = 13\\ 11*2 = 24\\ 11*3 = 35\\ 11*4 = 46\\ 11*5 = 57\\ 11*6 = 68\\ 11*7 = 79 \]

Identify the effect of multiplying an element by a constant on its standard deviation

When each element of a dataset is multiplied by a constant, the standard deviation of the dataset also gets multiplied by the absolute value of the same constant. In this case, the constant is 11.

Find the standard deviation of the second set of values

Since the standard deviation of the first dataset is 2 and we are multiplying each element in the first dataset by 11, the standard deviation of the second dataset will also get multiplied by 11: \[ \sigma(\text{second dataset}) = \sigma(\text{first dataset}) * 11 \] \[ \sigma(\text{second dataset}) = 2 * 11 = 22 \] So, the standard deviation of the second dataset is 22. Hence, the correct answer is (b) 22.

Key concepts

Understanding Standard Deviation in Statistics

Standard deviation is a term that often appears in the field of statistics. It is a measure of the amount of variation or dispersion in a set of values. In simpler words, it tells us how much the numbers in a data set differ from the average (mean) of the numbers.

For example, let's consider a set of numbers like the one in our exercise: 1, 2, 3, 4, 5, 6, and 7. Having a standard deviation of 2 means that, on average, the numbers in this set typically differ by 2 from their average value. The mathematical formula to calculate standard deviation is:\[ \sigma = \sqrt{\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n}} \], where \( \sigma \) is the standard deviation, \( n \) is the total number of values, \( x_i \) is each individual value, and \( \bar{x} \) is the mean of all values.

In our exercise, the standard deviation helps us understand the spread of the scores. It's a key statistic because a low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

Data Interpretation Through Statistical Analysis

Interpreting data is an essential aspect of statistics, as it allows one to understand and extract meaningful insights from numbers. The process often involves using statistical measures like mean and standard deviation to summarize data and make sense of it.

For instance, when we're given a set of numbers and we're told to find the standard deviation, we're not just performing a calculation; we're preparing to interpret what that standard deviation means for our dataset. In our standard deviation exercise, we use our understanding of the statistical measure to draw conclusions about how a set of scores is altered when each score is multiplied by a constant.

Effect of a Constant

When a constant multiplies every element in a dataset, as we saw with the number 11 in our exercise, the standard deviation is also multiplied by the absolute value of that constant. This direct proportionality is crucial in data interpretation, allowing us to predict how changes to the data will affect the spread of the values.

Mathematical Constants and Their Impact on Data

A mathematical constant is a fixed number that has a definite and unchanging value. Constants are the building blocks in mathematics that allow us to create formulae and equations that apply universally.

In the context of our exercise, the number 11 functions as a constant. This leads to an important property: whenever you multiply all data points by a constant, you affect certain statistics like the mean and standard deviation without changing others like the shape of the distribution.

The Role of Constants in Variation

The role of constants in statistics extends to understanding how data variation is influenced. The standard deviation, being a measure of this variation, is directly related to such constants. When we say that the standard deviation of a dataset gets multiplied by a constant, we’re touching on a key relationship in statistics. This property showcases how elegant and interconnected mathematical concepts are, and how they can be applied to real-world problems in data analysis and beyond.