The ages of 50 teachers are given below:
37 49 27 49 22 26 33 26 21 40
50 29 49 27 49 42 26 33 40 40
31 26 23 24 29 31 36 22 27 58
46 21 43 39 34 23 21 32 41 46
26 38 34 49 45 27 25 33 39 40
a. Form a frequency distribution table for the data using intervals of 21-25, 26-30, 31-35...
b. Calculate the mode, mean, and median.
c. Calculate the variance and standard deviation.
Answer (2)
The mode of the ages is the interval 26-30, the mean is approximately 36.96, and the median is 33. The variance is approximately 92.21, and the standard deviation is approximately 9.60.
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Create frequency distribution table: intervals of 21-25, 26-30, 31-35, 36-40, 41-45, 46-50, 51-55, 56-60 with corresponding frequencies.
Calculate the mode: the interval with the highest frequency is 26-30.
Calculate the mean: n ∑ x i = 34.367 .
Calculate the median: average of the 25th and 26th values after sorting, which is 33.
Calculate the variance: n − 1 ∑ ( x i − μ ) 2 = 94.779 .
Calculate the standard deviation: variance = 9.735 .
Mode: (26, 30), Mean: 34.367, Median: 33, Variance: 94.779, Standard Deviation: 9.735
Explanation
Problem Analysis We are given the ages of 50 teachers and asked to create a frequency distribution table, calculate the mode, mean, and median, and calculate the variance and standard deviation.
Frequency Distribution Table First, let's create the frequency distribution table using the given intervals. We count how many ages fall into each interval:
21-25: 21, 22, 23, 23, 21, 24, 22, 25, 21 (9)
26-30: 27, 26, 26, 29, 27, 26, 29, 27, 26, 27, 26 (11)
31-35: 33, 33, 31, 34, 31, 34, 33, 32 (8)
36-40: 37, 40, 40, 36, 39, 38, 39, 40 (8)
41-45: 42, 43, 41, 45 (4)
46-50: 49, 50, 49, 49, 49, 46, 46, 49 (8)
51-55: (0)
56-60: 58 (1)
Calculate the Mode Now, let's calculate the mode. The mode is the interval with the highest frequency. From the frequency distribution table, the interval 26-30 has the highest frequency (11).
Calculate the Mean Next, we calculate the mean. The mean is the average of all the ages. We sum all the ages and divide by the number of teachers (50):
Mean = n ∑ x i = 50 37 + 49 + 27 + ... + 39 + 40 = 50 1718.367 = 34.367
Calculate the Median To find the median, we first sort the data in ascending order. Then, since there are 50 data points (an even number), the median is the average of the 25th and 26th values.
Sorted data: 21, 21, 21, 22, 22, 23, 23, 24, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 29, 29, 31, 31, 32, 33, 33, 33, 34, 34, 36, 37, 38, 39, 39, 40, 40, 40, 40, 41, 42, 43, 45, 46, 46, 49, 49, 49, 49, 49, 50, 58
The 25th value is 33, and the 26th value is 33. Therefore, the median is (33 + 33) / 2 = 33.
Calculate the Variance and Standard Deviation Now, let's calculate the variance. The variance measures how spread out the data is from the mean.
Variance = n − 1 ∑ ( x i − μ ) 2
where μ is the mean (34.367) and n is the number of data points (50).
Variance = 49 ∑ ( x i − 34.367 ) 2 = 94.779
Finally, we calculate the standard deviation, which is the square root of the variance.
Standard Deviation = Variance = 94.779 = 9.735
Final Answer The frequency distribution table is:
21-25: 9
26-30: 11
31-35: 8
36-40: 8
41-45: 4
46-50: 8
51-55: 0
56-60: 1
The mode is the interval 26-30. The mean is 34.367. The median is 33. The variance is 94.779. The standard deviation is 9.735.
Examples
Understanding the distribution and central tendencies of teacher ages can be useful for policy makers in education. For instance, knowing the mean and median age can help in planning for retirement waves and recruiting new teachers. The variance and standard deviation provide insight into the age diversity within the teaching workforce, which can influence professional development strategies and mentorship programs. Frequency distribution helps to identify the concentration of teachers in specific age groups, which can be useful in resource allocation and training programs.
