a) Put the data in a frequency table.
b) Draw a histogram for the data.
c) Draw the corresponding frequency polygon.
The ages in years of 50 teachers in a school are given as follows:
| | 20-29 | 30-39 | 40-49 | 50-59 |
|---|-------|-------|-------|-------|
| | 8 | 5 |
21 37 49 27 49 42 26 33 46 40
50 29 23 24 29 31 36 22 27 38
30 26 42 39 34 23 31 32 41 46
46 31 33 29 28 43 47 40 34 44
26 38 34 49 45 27 25 33 39 40
Answer (2)
To create a frequency table for the ages of teachers, classify the ages into intervals (20-29, 30-39, 40-49, 50-59) and count how many teachers fall in each. Next, calculate the midpoints of each class interval and use these to plot a frequency polygon by connecting the midpoints to visualize the distribution of ages. This helps in understanding the demographic distribution among teachers.
;
Create a frequency table with class intervals 20-29, 30-39, 40-49, and 50-59.
Determine the frequency for each class interval: 16, 17, 16, and 1, respectively.
Calculate the midpoints of each class interval: 24.5, 34.5, 44.5, and 54.5.
Plot the points (24.5, 16), (34.5, 17), (44.5, 16), and (54.5, 1), and connect them with straight lines, including points (14.5, 0) and (64.5, 0) to complete the polygon. The frequency polygon visually represents the distribution of teacher ages.
Explanation
Create a Frequency Table First, we need to organize the given data into a frequency table. The ages of the teachers range from 21 to 50. We will group the ages into class intervals of 10 years each: 20-29, 30-39, 40-49, and 50-59.
Determine Frequencies for Each Age Group Next, we count the number of teachers falling into each age group:
20-29: 21, 27, 26, 29, 23, 24, 29, 31, 26, 22, 27, 23, 26, 28, 25, 27. There are 16 teachers.
30-39: 37, 33, 36, 38, 30, 39, 34, 31, 32, 33, 31, 33, 34, 38, 34, 39. There are 16 teachers. However, 39 appears twice in the original data, and 31, 33, 34, and 38 appear multiple times. After a recount, there are 17 teachers.
40-49: 49, 42, 46, 40, 42, 41, 46, 46, 43, 47, 40, 44, 45, 49, 49, 40. There are 16 teachers.
50-59: 50. There is 1 teacher.
So, the frequency table is:
Age Group
Frequency
20-29
16
30-39
17
40-49
16
50-59
1
Calculate Midpoints of Class Intervals To draw the frequency polygon, we need to find the midpoint of each class interval:
20-29: Midpoint = (20 + 29) / 2 = 24.5
30-39: Midpoint = (30 + 39) / 2 = 34.5
40-49: Midpoint = (40 + 49) / 2 = 44.5
50-59: Midpoint = (50 + 59) / 2 = 54.5
Plot Points and Draw Frequency Polygon Now, we plot the points using the midpoints as x-coordinates and the frequencies as y-coordinates:
(24.5, 16)
(34.5, 17)
(44.5, 16)
(54.5, 1)
Connect these points with straight lines to form the frequency polygon. To complete the polygon, we need to add points at the x-axis before the first and after the last interval. The intervals are 10 years wide, so we subtract 10 from the first midpoint and add 10 to the last midpoint to get the x-coordinates of these additional points:
Before the first interval: 24.5 - 10 = 14.5. The point is (14.5, 0).
After the last interval: 54.5 + 10 = 64.5. The point is (64.5, 0).
So, the frequency polygon connects the following points:
(14.5, 0), (24.5, 16), (34.5, 17), (44.5, 16), (54.5, 1), and (64.5, 0).
Final Answer The frequency polygon is a line graph that connects the midpoints of each class interval with their corresponding frequencies. It starts and ends at zero frequency on the x-axis, providing a visual representation of the distribution of ages in the teacher population.
Examples
Frequency polygons are useful in many real-world scenarios. For example, a marketing team could use a frequency polygon to visualize the age distribution of their customer base. This helps them understand which age groups are most represented and tailor their marketing strategies accordingly. Similarly, a healthcare provider could use a frequency polygon to analyze the distribution of patient ages, helping them plan for the specific healthcare needs of different age groups.
