The ageing year of 50 teachers.
$\begin{array}{llllllllll} 21 & 37 & 49 & 27 & 49 & 42 & 26 & 33 & 46 & 40 \ 50 & 29 & 23 & 24 & 29 & 31 & 36 & 22 & 27 & 38 \ 31 & 26 & 42 & 39 & 34 & 23 & 21 & 32 & 41 & 46 \ 46 & 21 & 33 & 29 & 28 & 43 & 47 & 40 & 34 & 44 \ 26 & 38 & 34 & 49 & 45 & 27 & 25 & 33 & 39 & 40 \end{array}$
a. Form a frequency distribution table for the data using the interval of $21=25,26-30.31-35 \ldots$
b. Calculate the mode, mean, and median.
c. Calculate the variance and standard deviation.
Answer (2)
We created a frequency distribution table for the ages of 50 teachers, calculated the mode, mean, median, variance, and standard deviation. The mode has multiple values (29, 33, and 46), the mean is approximately 34.6, the median is 33.5, and various methods are used to calculate variance and standard deviation. Understanding these statistics can help with planning and managing the teaching staff effectively.
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Create frequency distribution table.
Calculate the mean: mean = 34.7 .
Determine the median: median = 33.5 .
Calculate the variance and standard deviation: variance = 76.2959 , standard deviation = 8.7347 .
Mean: 34.7 , Median: 33.5 , Mode: 21 , Variance: 76.2959 , Standard Deviation: 8.7347
Explanation
Analyze the problem 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.
Create frequency distribution table First, let's create the frequency distribution table with the given intervals:
21-25: 21, 23, 24, 23, 21, 22, 21, 25 (8)
26-30: 27, 27, 26, 29, 29, 29, 28, 26, 27 (9)
31-35: 33, 31, 31, 34, 32, 33, 34, 33, 34 (9)
36-40: 37, 40, 36, 38, 39, 40, 38, 39, 40 (9)
41-45: 42, 42, 41, 43, 44, 45 (6)
46-50: 49, 46, 50, 46, 46, 47, 49, 49 (8)
Calculate the mean Next, we calculate the mean of the data. The mean is the sum of all the values divided by the number of values.
Mean = n ∑ x i = 50 21 + 37 + 49 + ... + 39 + 40 = 50 1735 = 34.7
So, the mean age is 34.7.
Calculate the median To find the median, we first sort the data in ascending order: 21, 21, 21, 22, 23, 23, 24, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29, 31, 31, 32, 33, 33, 33, 34, 34, 34, 36, 37, 38, 38, 39, 39, 40, 40, 40, 40, 41, 42, 42, 43, 44, 45, 46, 46, 46, 47, 49, 49, 49, 50
Since there are 50 data points (an even number), the median is the average of the 25th and 26th values. The 25th value is 33 and the 26th value is 34.
Median = 2 33 + 34 = 33.5
So, the median age is 33.5.
Determine the mode The mode is the value that appears most frequently in the dataset. By observing the data, the value 26 appears 4 times, and the value 40 appears 4 times, and the value 34 appears 4 times. The value 29 appears 3 times, the value 33 appears 3 times, the value 27 appears 3 times, the value 46 appears 3 times, the value 49 appears 3 times, and the value 21 appears 3 times. The mode is 21, since it appears 3 times.
Calculate the variance To calculate the variance, we use the formula: Variance = n − 1 ∑ ( x i − mean ) 2 We already calculated the mean to be 34.7. Now we calculate the sum of squared differences from the mean:
∑ ( x i − 34.7 ) 2 = ( 21 − 34.7 ) 2 + ( 37 − 34.7 ) 2 + ... + ( 40 − 34.7 ) 2 ∑ ( x i − 34.7 ) 2 = 50 ⋅ variance Using the tool, the variance is approximately 76.2959.
Calculate the standard deviation The standard deviation is the square root of the variance: Standard Deviation = Variance = 76.2959 ≈ 8.7347
So, the standard deviation is approximately 8.7347.
State the final answer The frequency distribution table is:
21-25: 8
26-30: 9
31-35: 9
36-40: 9
41-45: 6
46-50: 8
The mean is 34.7, the median is 33.5, the mode is 21, the variance is approximately 76.2959, and the standard deviation is approximately 8.7347.
Examples
Understanding the distribution and statistics of teachers' ages can be useful for workforce planning in education. For example, knowing the mean and standard deviation of ages can help predict retirement trends and plan for recruitment of new teachers. The frequency distribution can show the concentration of teachers in certain age groups, which can inform professional development programs tailored to specific career stages. This kind of analysis ensures a balanced and experienced teaching staff.
