The aging year has 50 teachers.
$\begin{array}{cccccccccc}
21 & 37 & 49 & 27 & 49 & 42 & 26 & 33 & 46 & 40 \
50 & 29 & 23 & 24 & 29 & 31 & 36 & 22 & 27 & 38 \
31 & 26 & 42 & 39 & 34 & 23 & 21 & 32 & 41 & 46 \
46 & 21 & 33 & 29 & 28 & 43 & 47 & 40 & 34 & 44 \
26 & 38 & 34 & 49 & 45 & 27 & 25 & 33 & 39 & 40
\end{array}$
a. Form a frequency distribution table partly done using the interval of 21-25, 26-30, 31-35...
b. Calculate the mode, mean, and median.
c. Calculate the variance and standard deviation.
Answer (2)
The frequency distribution table indicates that the most common age range among the teachers is 26-30. The calculations show that the mean age is 34.7, the median is 33.5, and the variance is 74.77 with a standard deviation of approximately 8.6469. These statistics provide insights into the age distribution of the teachers.
;
Create frequency distribution table: Group the ages into intervals and count the occurrences in each interval.
Calculate the mode: Identify the interval with the highest frequency.
Calculate the mean and median: Find the average age and the middle value of the sorted ages, respectively.
Calculate the variance and standard deviation: Measure the spread of the ages around the mean. Mode: 26 − 30 Mean: 34.7 Median: 34.0 Variance: 74.77 Standard Deviation: 8.647
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, we create the frequency distribution table with the given intervals:
21-25: Count the number of ages between 21 and 25 (inclusive). There are 8.
26-30: Count the number of ages between 26 and 30 (inclusive). There are 10.
31-35: Count the number of ages between 31 and 35 (inclusive). There are 9.
36-40: Count the number of ages between 36 and 40 (inclusive). There are 9.
41-45: Count the number of ages between 41 and 45 (inclusive). There are 6.
46-50: Count the number of ages between 46 and 50 (inclusive). There are 8.
So, the frequency distribution table is:
Interval
Frequency
21-25
8
26-30
10
31-35
9
36-40
9
41-45
6
46-50
8
Calculate mode, mean and median Next, we calculate the mode, mean, and median.
Mode: The mode is the interval with the highest frequency. In this case, the interval 26-30 has the highest frequency (10), so the mode is 26-30.
Mean: The mean is the average of all the ages. To calculate the mean, we sum all the ages and divide by the number of teachers (50).
Mean = 50 ∑ i = 1 50 x i
Mean = 50 21 + 37 + 49 + ... + 39 + 40 = 50 1735 = 34.7
Median: The median is the middle value when the ages are sorted in ascending order. Since there are 50 teachers (an even number), the median is the average of the 25th and 26th ages. First, we sort the data:
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
The 25th and 26th values are 33 and 34. Therefore,
$$\text{Median} = \frac{33 + 34}{2} = 33.5$$
Corrected Median: After sorting the data, the 25th value is 33 and the 26th value is 34. So the median is (33+34)/2 = 33.5
Calculate variance and standard deviation Finally, we calculate the variance and standard deviation.
Variance: The variance is the average of the squared differences from the mean.
Variance = 50 ∑ i = 1 50 ( x i − Mean ) 2
Variance = 50 ∑ i = 1 50 ( x i − 34.7 ) 2 = 74.77
Standard Deviation: The standard deviation is the square root of the variance.
Standard Deviation = Variance = 74.77 = 8.646964785403025
State the final answer The frequency distribution table is:
Interval
Frequency
21-25
8
26-30
10
31-35
9
36-40
9
41-45
6
46-50
8
The mode is 26-30, the mean is 34.7, the median is 34.0, the variance is 74.77, and the standard deviation is 8.646964785403025.
Examples
Understanding the distribution of ages in a population, such as teachers in a school district, can be useful for resource allocation and planning. For example, if a large proportion of teachers are nearing retirement age, the district may need to focus on recruiting and training new teachers. Similarly, knowing the mean and standard deviation of teacher ages can help in understanding the overall experience level and diversity within the teaching staff. These statistical measures provide valuable insights for effective management and policy-making in educational institutions.
