Question 44·Medium·One-Variable Data Distributions; Measures of Center and Spread
The frequency table summarizes the 25 data values in a data set.
| Data value | Frequency |
|---|---|
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 4 |
| 7 | 3 |
| 8 | 3 |
| 9 | 1 |
What is the median of the data set?
(Express the answer as an integer)
For median questions from a frequency table, first add the frequencies to find the total number of data values. If the total is odd, compute the position of the median using ; if it is even, the median is the average of the values at positions and . Then run a cumulative total down the frequency column to see which data value contains that position. This avoids writing out the full list and is faster and less error-prone on test day.
Hints
Think about the definition of median
The median is the middle value when the data are ordered. First, use the frequency table to figure out how many total data values there are.
Use the total number of data values
Once you know there are 25 values, remember that for an odd number of data points, the median is at position in the ordered list. What position does that give here?
Locate that position in the table
Use cumulative (running) totals of the frequencies to see which data value covers the position you found. For example, values of 2 occupy positions 1–2, values of 3 occupy positions 3–5, and so on. Which data value includes the median position?
Desmos Guide
Enter the full data set as a list
In a new expression line, type the data values, repeating each number according to its frequency, inside curly braces. For example: {2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,8,9} (all on one line). This represents all 25 data values.
Use Desmos to compute the median
In another expression line, type median(L1) if Desmos named your list L1, or median({2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,8,8,8,9}) directly. Read the number Desmos outputs; that is the median of the data set.
Step-by-step Explanation
Recall what the median is and find how many values there are
The median is the middle value when all data are listed in order from least to greatest.
Use the frequency column to find how many data values are in the set:
- For value 2, there are 2 data points.
- For value 3, there are 3 data points.
- For value 4, there are 4 data points.
- For value 5, there are 5 data points.
- For value 6, there are 4 data points.
- For value 7, there are 3 data points.
- For value 8, there are 3 data points.
- For value 9, there is 1 data point.
Add the frequencies:
So there are data values in total.
Find the position of the median
When there are data values and is odd, the median is the value at position
in the ordered list.
Here , so the median is at position
So we need to find the 13th value in the ordered data set.
Use cumulative frequencies to locate the 13th value
Now use the frequencies to see which data value is in position 13 when the data are listed from smallest to largest.
Compute cumulative (running) totals of the frequencies:
- Value 2: positions 1 through 2 (2 values so far)
- Value 3: positions 3 through 5 (2+3 = 5 values so far)
- Value 4: positions 6 through 9 (5+4 = 9 values so far)
The next group of equal values occupies positions 10 through 14, which includes position 13. Identify which data value this group corresponds to using the table.
State the median
Since the 13th value in the ordered list is 5, the median of the data set is