The ages of $n$ teachers are:
$\begin{array}{llllllllll}
21 & 37 & 49 & 27 & 49 & 22 & 26 & 33 & 46 & 40 \
50 & 29 & 23 & 24 & 29 & 31 & 36 & 22 & 27 & 58 \
31 & 26 & 42 & 39 & 34 & 23 & 21 & 32 & 41 & 46 \
46 & 21 & 33 & 29 & 28 & 43 & 47 & 40 & 34 & 44 \
26 & 38 & 34 & 49 & 45 & 27 & 25 & 33 & 39 & 40
\end{array}$
a. Form a frequency distribution table with 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)
We created a frequency distribution table for the ages of 50 teachers, calculated the mode as 21, 29, 46, 49; the mean as 34.8; the median as 32.5. Additionally, the variance is 52.5 and the standard deviation is approximately 7.25.
;
Create frequency distribution table.
Calculate the mean: 50 1740 = 34.8 .
Calculate the median: 2 32 + 33 = 32.5 .
Calculate the variance and standard deviation: Variance = 101.53, Standard Deviation = 10.08. M e an = 34.8 , M e d ian = 32.5 , Va r ian ce = 101.53 , St an d a r d De v ia t i o n = 10.08
Explanation
Understand the problem and provided data We are given a dataset of ages of n teachers. The data is presented in a table with 5 rows and 10 columns, so there are a total of n = 50 teachers. The data values are: 21, 37, 49, 27, 49, 22, 26, 33, 46, 40, 50, 29, 23, 24, 29, 31, 36, 22, 27, 58, 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. We need to: a) Form a frequency distribution table using intervals of 21-25, 26-30, 31-35, ... b) Calculate the mode, mean, and median. c) Calculate the variance and standard deviation.
Create the frequency distribution table First, let's create the frequency distribution table. The intervals are: 21-25, 26-30, 31-35, 36-40, 41-45, 46-50, 51-55, 56-60. We count the number of data points that fall into each interval:
21-25: 21, 22, 23, 24, 25, 21, 23 = 7
26-30: 27, 29, 26, 29, 27, 28, 26, 29 = 8
31-35: 33, 31, 34, 31, 32, 33, 34, 34 = 8
36-40: 37, 40, 36, 39, 38, 40, 39, 40 = 8
41-45: 42, 41, 43, 44, 45 = 5
46-50: 49, 46, 50, 46, 46, 47, 49, 49 = 8
51-55: None = 0
56-60: 58 = 1
Calculate the mean Now, let's calculate the mean. The formula for the mean is: Mean = n ∑ i = 1 n x i .
Sum of all data points = 21+37+49+27+49+22+26+33+46+40+50+29+23+24+29+31+36+22+27+58+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 = 1740
Mean = 50 1740 = 34.8
Calculate the mode Next, let's calculate the mode. The mode is the value that appears most frequently in the dataset. By observing the data, we can see that the values 26, 27, 29, 33, 34, 40, 46, and 49 appear three times each. Therefore, the modes are 26, 27, 29, 33, 34, 40, 46, and 49.
Calculate the median Now, let's calculate the median. First, we need to arrange the data in ascending order:
21, 21, 21, 22, 22, 23, 23, 24, 25, 26, 26, 26, 26, 27, 27, 27, 27, 28, 29, 29, 29, 29, 31, 31, 32, 33, 33, 33, 33, 34, 34, 34, 34, 36, 37, 38, 39, 39, 40, 40, 40, 40, 41, 42, 43, 44, 45, 46, 46, 46, 46, 47, 49, 49, 49, 50, 58
Since there are 50 data points (an even number), the median is the average of the 25th and 26th values. The 25th value is 32, and the 26th value is 33.
Median = 2 32 + 33 = 32.5
Calculate the variance Now, let's calculate the variance. The formula for the variance is: Variance = n − 1 ∑ i = 1 n ( x i − Mean ) 2 .
We already know that the mean is 34.8. We need to calculate the sum of squared differences from the mean:
∑ i = 1 n ( x i − Mean ) 2 = ( 21 − 34.8 ) 2 + ( 37 − 34.8 ) 2 + ... + ( 40 − 34.8 ) 2
Calculating each term and summing them up, we get: ∑ i = 1 n ( x i − Mean ) 2 = 4975.2
Variance = 50 − 1 4975.2 = 49 4975.2 = 101.534693877551
Calculate the standard deviation Finally, let's calculate the standard deviation. The formula for the standard deviation is: Standard Deviation = Variance .
Standard Deviation = 101.534693877551 = 10.07644256
Final Answer a) Frequency Distribution Table:
21-25: 7
26-30: 8
31-35: 8
36-40: 8
41-45: 5
46-50: 8
51-55: 0
56-60: 1
b) Mode: 26, 27, 29, 33, 34, 40, 46, 49
Mean: 34.8
Median: 32.5
c) Variance: 101.53
Standard Deviation: 10.08
Examples
Understanding the distribution and statistics of teacher ages can be useful for school administrators in planning professional development programs, retirement projections, and recruitment strategies. For instance, knowing the mean and standard deviation of teacher ages can help in predicting future staffing needs and ensuring a balanced distribution of experience levels within the teaching staff. The frequency distribution can highlight age clusters, aiding in targeted interventions or support programs. Calculating these statistics provides a quantitative basis for making informed decisions about human resource management in educational institutions.
