Let us delve into the fascinating world of statistics and take a look at what the median is in statistics, along with its examples, uses, and types.
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The median is a measure of central tendency in statistics that indicates the middle value of a dataset. It is the value that divides the dataset into two equal portions, with half of the values above and the rest below the median.
The median is computed by sorting the data from smallest to biggest (or greatest to smallest) and determining the midpoint value.
If the number of observations in the dataset is odd, the median is the midway value. If the number of observations in the collection is even, the median is the average of the two middle values. Since it is unaffected by extreme values or outliers in the sample, the median is a helpful indicator of central tendency.
Mentioned below are examples of finding the median in statistics for both odd and even numbers of observations:
Case 1: The Number of Observations is Odd.
- Order the dataset from smallest to largest (or vice versa).
- The median of this dataset would be the middle value. This can be found by identifying the value that lies at the (n+1)/2th position, where n is the total number of observations.
For example, consider the following dataset of exam scores:
70, 85, 60, 90, 80, 75, 95
Finding the median
Order the dataset from smallest to largest:
60, 70, 75, 80, 85, 90, 95
Since the dataset has 7 observations (an odd number), the median is the (7+1)/2 = 4th observation, which is 80.
Case 2: The Number of Observations is Even.
- Order the dataset from smallest to largest (or vice versa).
- The median of this dataset will be the average of the two middle values. This can be found by identifying the values that lie at the n/2th and (n/2)+1th positions, where n is the total number of observations.
For example, consider the following dataset of exam scores:
70, 85, 60, 90, 80, 75
Finding the median
Order the dataset from smallest to largest:
60, 70, 75, 80, 85, 90
Since the dataset has 6 observations (an even number), the median is the average of the two middle values, which are the 3rd and 4th observations. Therefore, the median is (75 + 80)/2 = 77.5.
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In statistics, aside from being a nice summary statistic, the median has several essential qualities that make it a useful tool in data analysis. Mentioned below are some essential qualities of the median in statistics:
- Robustness: An essential quality of the median is its robustness. Unlike the mean, which is heavily influenced by outliers or extreme values in the sample, the median is essentially unaffected by such values. This gives it a more robust measure of central tendency when the dataset contains extreme values.
- Unique Value: Whether the dataset comprises an even or odd number of observations, the median always corresponds to a unique value: with an odd number of observations, the median is the middle value, and with an even number of observations, the median is the average of the two middle values.
- Equal Parts: The median divides the dataset into two equal portions, with half of the observations above and the rest below the median. For skewed distributions or datasets containing outliers, this characteristic makes the median a useful estimate of central tendency.
- Change Resistance: Another essential feature of the median is its resistance to changes in the dataset that do not alter the middle value. Adding or subtracting observations from the ends of the collection, for example, has no effect on the median as long as the center value remains constant.
- The Median is Unaffected by Outliers: The median remains unaffected by the outliers in the dataset. As a result, it is a good measure of the central tendency for datasets with a lot of outliers or for skewed distributions.
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A median class in statistics is a class interval in a frequency distribution that contains the distribution’s median.
A frequency distribution is a method of categorizing data and counting the number of observations that fall into each interval (or class). The median of a frequency distribution is the value that divides the lower 50% of observations from the higher 50%.
When calculating the median of a frequency distribution, we need to identify the median class, or the class interval that contains the median value. To find the median class, we first calculate the cumulative frequency of each class interval, i.e., the sum of the frequencies up to and including that interval. We then find the class interval where the cumulative frequency first exceeds half of the total frequency, i.e., the median of the distribution.
| Class Interval |
Frequency |
| 50-59 |
5 |
| 60-69 |
12 |
| 70-79 |
20 |
| 80-89 |
15 |
| 90-99 |
8 |
To find the median class, we first calculate the cumulative frequencies, as shown below:
| Class Interval |
Frequency |
Cumulative Frequency |
| 50-59 |
5 |
5 |
| 60-69 |
12 |
17 |
| 70-79 |
20+ |
37 |
| 80-89 |
15 |
52 |
| 90-99 |
8 |
60 |
The total frequency of the dataset above is 60, so half of the total frequency is 30. The median class is the class interval where the cumulative frequency first exceeds 30. In this case, the median class is the interval 70–79 since the cumulative frequency at the end of this interval (37) exceeds 30.
Having identified the median class, we can then use it to calculate an estimated median value for the frequency distribution. Using the below-mentioned formula is one way to tackle this:
Median = L + ((n/2 - CF)/f) x w
where L is the lower boundary of the median class, n is the total frequency, CF is the cumulative frequency up to the end of the median class, f is the frequency of the median class, and w is the width of each class interval.
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Medians are beneficial since they are not impacted in the same manner as means by extreme numbers or outliers. In statistics, numerous forms of medians are often used, some of which are mentioned below:
- Simple Median: The simple median is derived by sorting the data and finding the middle number. In cases with an odd number of data points, the median would simply be the middle value, whereas with an even number of data points, the median would be the average of the two values in the middle.
- Weighted Median: Not all data points are equally relevant in some circumstances, and a weighted median can be used to lend additional weight to specific data points. To compute a weighted median, multiply each data point by its weight, and then compute the median using the weighted data.
- Running Median: A running median is a type of median calculated for a changing range of data points. For studying time series data or other data that varies over time, this is useful. A running median can assist in identifying trends or patterns in data.
- Trimmed Median: A trimmed median is a sort of median that is produced by deleting a specific percentage of data points from the dataset’s top and bottom. This is beneficial when the data has extreme values or outliers that distort the results.
- Hodges–Lehmann Median: The Hodges-Lehmann median is a robust measure of central tendency that is obtained by calculating the median of all potential pairwise averages of the data points. This strategy is beneficial when the data is not regularly distributed or contains outliers.
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Conclusion
In statistics, the median is an essential measure of central tendency that indicates the midpoint value of a dataset. The median, unlike the mean, is unaffected by extreme values or outliers, which makes it a reliable statistic in many applications. Depending on the properties of the data, other types of medians can be used, such as the simple median, weighted median, running median, trimmed median, and Hodges-Lehmann median.
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