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.
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.