To determine the first quartile (Q1), we need to first calculate the total number of data points. We can do this by adding up the frequencies:
3 + 5 + 9 + 11 + 2 = 30
Since Q1 is the value below which 25% of the data falls, we need to find the cumulative frequency (CF) that corresponds to the 25th percentile.
25% of 30 is 7.5, which means that Q1 corresponds to the 7th data point. Looking at the frequency table, we can see that the 7th data point falls within the [97; 107) class.
To interpolate the exact value of Q1 within the [97; 107) class, we can use the formula:
Q1 = L + [(N/4 - CF) / f] * w
Where:
- L = lower boundary of the class interval containing Q1 (97 in this case)
- N = total number of data points (30)
- CF = cumulative frequency of the class interval before Q1 (3 in this case)
- f = frequency of the class interval containing Q1 (5 in this case)
- w = width of the class interval (10)
Plugging in the values:
Q1 = 97 + [(30/4 - 3) / 5] * 10
Q1 = 97 + [(7.5 - 3) / 5] * 10
Q1 = 97 + (4.5 / 5) * 10
Q1 = 97 + 0.9 * 10
Q1 = 97 + 9
Q1 = 106
Therefore, the first quartile (Q1) for this data set is 106.
Consider the following partially completed frequency table:
Class xi fi
[87; 97) 92 3
[97; 107) 102 5
[107; 117) 112 9
[117; 127) 122 11
[127; 137) 132 2
Determine the first quartile by using the appropriate formula.
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