The median and the mode of the following wage distribution are known to be 33.5 and 34 respectively. Three frequency values from the table are however, missing total frequency 230
Class interval 0- 10- 20- 30- 40- 50- 60-
10 20 30 40 50 60 70
Frequency 4 16 f1 f2 f3 4 6
INSTRUCTION A. determine missing frequency B. Determine mean
C. Draw histogram
To determine the missing frequencies, we can use the information given about the total frequency and the values for the median and mode.
A. Determine missing frequency:
We know that the total frequency is 230, and we can calculate the sum of the frequencies given in the table:
4 + 16 + f1 + f2 + f3 + 4 + 6 = 30 + f1 + f2 + f3 = 30 + (f1 + f2 + f3)
We also know that the mode is 34, so the frequency with the highest value is 34. Looking at the table, the class interval with this frequency is 30-40. Therefore, the missing frequency f2 = 34 - 4 = 30.
Now, we can use the median to determine the values of f1 and f3.
The median is the middle value of the dataset. In this case, the median is 33.5, which lies in the 30-40 class interval. Since the median falls in the lower half of the interval, the frequencies for the class intervals 0-10, 10-20, 20-30, and 30-40 will add up to half of the total frequency. Let's calculate this:
16 + f1 + f2 = (230/2)
16 + f1 + 30 = 115
f1 + 46 = 115
f1 = 115 - 46
f1 = 69
To find f3, we subtract the frequencies we've already determined from the total frequency:
Total frequency = 230
f3 = Total frequency - (f1 + f2 + 4 + 6)
f3 = 230 - (69 + 30 + 4 + 6)
f3 = 230 - 109
f3 = 121
Therefore, the missing frequencies are f1 = 69, f2 = 30, and f3 = 121.
B. Determine mean:
The mean is calculated by summing up all the values multiplied by their respective frequencies and then dividing by the total frequency. We can use the midpoint of the class intervals as the values.
Mean = (4*(5) + 16*(15) + 69*(25) + 30*(35) + 121*(45) + 4*(55) + 6*(65)) / 230
Mean = (20 + 240 + 1725 + 1050 + 5445 + 220 + 390) / 230
Mean = 11090 / 230
Mean = 48.2174 (rounded to four decimal places)
Therefore, the mean is approximately 48.2174.
C. Draw histogram:
A histogram is a graphical representation of a frequency distribution, where the classes are represented by bars.
To draw the histogram, we can place the class intervals on the x-axis, and the frequency on the y-axis. The height of each bar will represent the frequency for the respective class interval.
Here is a representation of the histogram for the given frequency distribution:
|
|
|
30-40 | 30 ||
|
|
20-30 | 16 ||
|
| | 6 |
10-20 || 4 ||
|
| | 4 |
0-10 || 4 ||
Note: The bars should be properly labeled and proportional to the frequencies.