For data -1, 1, 4, 3, 8, 12, 17, 19, 9, 11; if M is the median of first 5 observations and N is the median of last five observations, then what is the value of 4M - N ?
0
Calculating Median and Evaluating an Expression
The problem asks us to find the value of the expression \(4M - N\), where \(M\) is the median of the first five observations and \(N\) is the median of the last five observations from a given dataset.
The given data set is: -1, 1, 4, 3, 8, 12, 17, 19, 9, 11.
Finding the Median of the First 5 Observations (M)
The first 5 observations are: -1, 1, 4, 3, 8.
To find the median, we must first arrange these observations in ascending order.
- Original: -1, 1, 4, 3, 8
- Sorted: -1, 1, 3, 4, 8
Since there are 5 observations (an odd number), the median is the middle value in the sorted list. The middle value is the 3rd observation.
Therefore, M = 3.
Finding the Median of the Last 5 Observations (N)
The last 5 observations are: 12, 17, 19, 9, 11.
To find the median, we must first arrange these observations in ascending order.
- Original: 12, 17, 19, 9, 11
- Sorted: 9, 11, 12, 17, 19
Since there are 5 observations (an odd number), the median is the middle value in the sorted list. The middle value is the 3rd observation.
Therefore, N = 12.
Calculating the Expression 4M - N
Now we need to evaluate the expression \(4M - N\) using the values of M and N we found.
- M = 3
- N = 12
Substitute these values into the expression:
\(4M - N = 4 \times 3 - 12\)
Perform the multiplication:
\(4 \times 3 = 12\)
Now substitute this back into the expression:
\(12 - 12\)
Perform the subtraction:
\(12 - 12 = 0\)
So, the value of \(4M - N\) is 0.
Let's summarize the steps:
- Identify the first 5 observations and find their median (M).
- Identify the last 5 observations and find their median (N).
- Calculate the value of \(4M - N\).
The calculation confirms the result is 0.
Revision Table: Key Concepts
| Concept | Description | How to Calculate (for a list of numbers) |
|---|---|---|
| Median | The middle value in a dataset that is ordered from least to greatest. It is a measure of central tendency. | 1. Arrange data in ascending order. 2. If number of observations (n) is odd, median is the \(\left(\frac{n+1}{2}\right)^{\text{th}}\) value. 3. If number of observations (n) is even, median is the average of the \(\left(\frac{n}{2}\right)^{\text{th}}\) and \(\left(\frac{n}{2}+1\right)^{\text{th}}\) values. |
Additional Information: Data Analysis Terms
Understanding terms like median, mean, and mode is fundamental in data analysis and statistics. These are all measures of central tendency, helping us understand the typical value in a dataset.
- Mean: The average of all the numbers in a dataset. Calculated by summing all values and dividing by the number of values.
- Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode (if all values appear with the same frequency).
- Observations: Individual data points or values recorded in a dataset.
Finding the median requires ordering the data, which is a key step in understanding the distribution of values.
Important Questions from Statistics
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1, 2, 1, 1, 2, 1, 4, 6, 5, 4
What will be the mode of data so obtained ?
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x 1 2 3 5 f 4 6 9 7 What is the value of median of the distribution ?
Let P, Q, R represent mean, median and mode. If for some distribution \(5 P=4 Q=\frac{R}{2}\) then what is \(\frac{P+Q}{2 P+0.7 R}\) equal to ?
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