If the yield (in gm) of barley from 7 plots of size one square yard each, were found to be 180, 191, 175, 111, 154, 141 and 176 then what is the median yield?
175 gm
Finding the Median Yield of Barley
The question asks us to find the median yield of barley from a given set of yield measurements. The yield is given in grams (gm) for 7 plots of size one square yard each.
What is the Median?
The median is a measure of central tendency. It represents the middle value in a dataset when the data is arranged in ascending or descending order. It is useful because it is not affected by extremely large or small values (outliers) as much as the mean is.
Steps to Find the Median Yield
To find the median of a dataset, we follow these steps:
- Arrange the data points in ascending order (from smallest to largest).
- Determine the number of data points, denoted by \(n\).
- If \(n\) is odd, the median is the middle value. Its position is given by the formula \(\frac{n+1}{2}\).
- If \(n\) is even, the median is the average of the two middle values. The positions of these two values are \(\frac{n}{2}\) and \(\frac{n}{2} + 1\).
Applying the Steps to the Barley Yield Data
The given yields of barley from 7 plots are: 180, 191, 175, 111, 154, 141, and 176 gm.
Step 1: Arrange the data in ascending order.
Let's list the yields and then sort them:
- Original data: 180, 191, 175, 111, 154, 141, 176
- Ordered data: 111, 141, 154, 175, 176, 180, 191
Step 2: Determine the number of data points (\(n\)).
There are 7 yield values given. So, \(n = 7\).
Step 3: Find the median position.
Since \(n=7\) is an odd number, the median is the single middle value. The position of the median is:
\(\text{Median position} = \frac{n+1}{2} = \frac{7+1}{2} = \frac{8}{2} = 4\)
The median is the 4th value in the ordered list.
Step 4: Identify the median value.
The ordered data is: 111, 141, 154, 175, 176, 180, 191.
The 4th value in this ordered list is 175.
Therefore, the median yield is 175 gm.
This means that half of the barley yields are less than or equal to 175 gm, and half are greater than or equal to 175 gm.
The calculated median yield is 175 gm.
Revision Table: Understanding Median Calculation
| Concept | Description | How to Find |
|---|---|---|
| Median | The middle value of a dataset when ordered. | Order data, find middle value(s). |
| Odd \(n\) | Number of data points is odd. | Median is the value at position \(\frac{n+1}{2}\). |
| Even \(n\) | Number of data points is even. | Median is the average of values at positions \(\frac{n}{2}\) and \(\frac{n}{2} + 1\). |
| Barley Yield Data | 111, 141, 154, 175, 176, 180, 191 | \(n=7\) (odd), Median is the 4th value = 175 gm. |
Additional Information: Measures of Central Tendency
The median is one of several measures used to describe the center of a dataset. Other common measures of central tendency include:
- Mean (Average): The sum of all values divided by the number of values. It is sensitive to extreme values.
- Mode: The value that appears most frequently in the dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode.
Choosing the appropriate measure of central tendency depends on the nature of the data and the purpose of the analysis. For skewed data or data with outliers, the median is often a better indicator of the typical value than the mean, as it is less affected by these extreme points.
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