Statistical Tools and Interpretation Test - 4

Arranging the data in ascending order, 

\(10,15,18,20,21,25,28,30,35,36,40 \)

\({N}=11\)

Median \(=\) size of \(\left(\frac{N+1}{2}\right)^{\text {th }}\) item

or, Median \(=\) size of \(\left(\frac{11+1}{2}\right)^{\text {th }}\) item

Thus, Median is given by the size of \(6^{\text {th }}\) item. 

The sixth item in the data \(10,15,18,20,21,25,28,30,35,36,40= 25\)

So, Median of the data so given is 25.

For a nominal scale that identifies users versus non-users of bank credit cards, the appropriate measure of central tendency is the mode. The mode represents the category (in this case, "users" or "non-users") that occurs most frequently in the dataset. It indicates the most common response or category among the observations.

Arranging the data in an ascending order,

\(11,12,14,18,22,26,30,32,35,41\)

\(Q_1=\) size of \(\frac{(\mathrm{N}+1)^{\text {th }}}{4}\) item 

\(=\) size of \(\frac{(10+1)^{\text {th }}}{4}\) item \(=\) size of \(2.75^{\text {th }}\) item

\(=2 \text {nd item }+.75 (3 \text {rd item }-2 \text { nd item }) \)

\(=12+.75(14-12)=13.5\) marks

The median is often the preferred measure of central tendency if the data are severely skewed. Skewed data sets have a distribution where the values are not evenly distributed around the center but are instead concentrated more heavily on one side. In skewed data sets, the median is often preferred because it gives a more robust estimate of the central tendency that is less influenced by outliers.

Converting the given table into frequency distribution table:

The mean is the average of the numbers.

So if \(x=\) all 11 numbers added together, then 

\(\frac{x}{11}=7\) 

So \(x=77\) (this is all of the 11 numbers added together)

If the number 13 is removed, the equation becomes:

\(\frac{(77-13)}{10}=M \quad(M\) is the new mean \()\)

\(M = \frac{64}{10}=6.4\)

So, the new mean is 6.4

Sum of the incorrect observations \(=45 \times 100=4500\)

Sum o the correct observations \(=4500-(91+13)+(19+31) = 4446\)

Mean = Sum of all the terms/Total number of terms

Correct mean \(=\frac{4446}{100}=44.46\)

To compute the median, the dataset is first sorted from smallest to largest or vice versa. Then, if the dataset has an odd number of values, the median is the middle value. If the dataset has an even number of values, the median is the average of the two middle values. This process ensures that the median accurately represents the central tendency of the data, making it a robust measure even in the presence of extreme values.

Class interval width (c) \(=120-100=20\)

Let assumed mean \(A=150\)

The 82nd percentile means that the student's performance is better than 82% of the total candidates. Since one lakh students appeared, 82% of one lakh is 82,000. Therefore, the student stands below 18,000 students (100,000 - 82,000) in terms of their performance compared to others.