Statistical Tools and Interpretation Test - 5

The third quartile (Q₃) represents the point in a dataset where 75% of the observations fall below it and 25% of the observations are above it. In a dataset of 100 observations, the 75th percentile corresponds to Q₃, indicating that 75 observations are below this value and 25 observations are above it.

When each observation is increased by \(10 \%\), it means each observation is multiplied by \(1+\frac{10}{100}=1.1\).

So, if the mean of the original observations is 10 , after increasing each observation by \(10 \%\), the new mean will also increase by the same factor.

Therefore, the new mean will be \(10 \times 1.1=11\).

First, we will calculate the sum of all the data points multiplied by their respective frequencies i.e. ∑fx 

Now, we know that, 

Arithmetic mean = \(\frac{\sum fx}{\sum f}= \frac{1015}{45}=22.55\)

As we know,

\(\Rightarrow\) Mean - Mode =3 Mean - Median

\(\Rightarrow\) Mean- Mode =3 Mean -3 Median

\(\Rightarrow\) 3 Median - Mode = 2 Mean

\(\Rightarrow\) \(\frac{1}{2}\)(3 Median -  Mode) = Mean

\(\therefore\) \(\mathrm{k}=\frac{1}{2}\)

Standard deviation is not a measure of central tendency. A measure of central tendency is a typical value for a probability distribution. The measures of central tendency, such as the mean, median, and mode, are used to describe the center or typical value of a dataset.

Standard deviation measures the dispersion or spread of a dataset around the mean. It quantifies the average distance of data points from the mean. It provides information about the variability or scatter of the data points.

The first ' \(n\) ' natural numbers are \(1,2,3, \ldots . ., n\) and their corresponding weights are \(1,2,3, \ldots, n\)

\(\therefore\) Weighted Mean\(=\overline{\mathrm{x}}_{\mathrm{w}}=\frac{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{w}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{w}_{\mathrm{i}}}\)

\(\Rightarrow \frac{1 \times 1+2 \times 2+3 \times 3+ \ldots \ldots+\mathrm{n} \times \mathrm{n}}{1+2+3+\ldots \ldots+\mathrm{n}}\)

\(\Rightarrow \frac{1^2+2^2+3^3+\ldots \ldots .+\mathrm{n}^2}{1+2+3+\ldots+\mathrm{n}}\)

As sum of square of \(n\) natural numbers is \(=\frac{n(n+1)(2 n+1)}{6}\)

\(\therefore\) mean \(=\frac{\frac{1}{6} \mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{\frac{1}{2} \mathrm{n}(\mathrm{n}+1)}\)

\(\therefore\) mean \(=\frac{1}{3}(2 \mathrm{n}+1)\)

Mode = No. of element which is repeated more.

Mode \(=5\) and 9

\(\because\) Both are repeated 3 times in the series.

Median is the middle most value of a given series that represents the whole class of the series. So since it is a positional average, it is calculated by observation of a series and not through the extreme values of the series which. Therefore, median is not affected by the extreme values of a series.

If the average of a set of seven numbers is 8 , then the sum of these seven numbers is

\(7 \times 8=56\)

When the eighth number is added to the set, making it a total of eight numbers, and the average remains 8 , it means the sum of the eight numbers is still \(8 \times 8=64\).

Number added \(=\) Sum of eight numbers - Sum of seven original numbers

Number added \(=64-56\)

So, the number added \(=8\)

In a symmetric distribution, such as a normal distribution, the median (Q₂) is the midpoint of the dataset. If the median is 50, it means that 50% of the observations are below 50 and 50% are above 50. Similarly, in a symmetric distribution, the first quartile (Q₁) would be the midpoint of the lower 50% of the data, which is 25, and the third quartile (Q₃) would be the midpoint of the upper 50% of the data, which is 75. Therefore, the likely range of values for Q₁ and Q₃ in this case would be Q₁ = 25 and Q₃ = 75.