Finding the median from a histogram involves approximating the value that divides the data set into two equal halves. Since you don't have the raw data, you estimate the median's location within a bin. Here's how to do it:
1. Determine the Total Frequency (n):
- Sum the frequencies of all the bars in the histogram. This represents the total number of data points.
2. Calculate the Median Position:
- The median is the value that splits the data in half. Its position is approximately at
(n + 1) / 2. Ifnis even, the median is often taken as the average of the values at positionsn/2and(n/2) + 1. For histograms with grouped data, we typically use a single approximation.
3. Identify the Median Bin:
- Starting from the left side of the histogram, add up the frequencies of each bin until you reach or exceed the median position calculated in step 2. The bin where you reach or exceed the median position is the median bin.
4. Interpolate Within the Median Bin:
-
This is where the approximation comes in. We assume the data within the median bin is uniformly distributed. Use the following formula to estimate the median:
Median = L + (((n/2) - CF) / f) * wWhere:
L= Lower boundary of the median bin.n= Total frequency (from step 1).CF= Cumulative frequency of the bins before the median bin.f= Frequency of the median bin.w= Width of the median bin (the bin size).
Example:
Let's say we have the following histogram data:
| Bin | Frequency |
|---|---|
| 10-20 | 5 |
| 20-30 | 12 |
| 30-40 | 18 |
| 40-50 | 10 |
| 50-60 | 5 |
-
Total Frequency (n): 5 + 12 + 18 + 10 + 5 = 50
-
Median Position: (50 + 1) / 2 = 25.5
-
Median Bin:
- The first bin (10-20) has a frequency of 5.
- The first two bins (10-20 and 20-30) have a combined frequency of 5 + 12 = 17.
- The first three bins (10-20, 20-30, and 30-40) have a combined frequency of 5 + 12 + 18 = 35. Since 35 exceeds 25.5, the median bin is 30-40.
-
Interpolation:
- L = 30 (lower boundary of the median bin)
- n = 50
- CF = 17 (cumulative frequency before the median bin: 5 + 12)
- f = 18 (frequency of the median bin)
- w = 10 (width of the median bin: 40 - 30)
Median = 30 + (((50/2) - 17) / 18) * 10
Median = 30 + ((25 - 17) / 18) * 10
Median = 30 + (8 / 18) * 10
Median = 30 + 4.44
Median ≈ 34.44
Therefore, the approximate median for this data, based on the histogram, is 34.44.
In summary, find the total frequency, determine the median's position, identify the bin containing the median, and then use interpolation to approximate the median's value within that bin. Remember this method provides an estimate due to the grouped nature of histogram data.
