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$\begin{array}{cccccccccc}
21 & 37 & 49 & 27 & 49 & 42 & 26 & 33 & 246 & 40 \\
50 & 29 & 23 & 24 & 29 & 31 & 36 & 22 & 27 & 38 \\
31 & 26 & 42 & 39 & 34 & 23 & 21 & 32 & 41 & 46 \\
46 & 21 & 33 & 29 & 28 & 43 & 47 & 40 & 34 & 44 \\
26 & 38 & 34 & 49 & 45 & 27 & 25 & 33 & 39 & 40
\end{array}$
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)
The frequency distribution table for the data indicates the intervals and their counts. The mode is 26-30, the mean is approximately 34.36, while the median is 33.5. Finally, the variance is approximately 74.77 and the standard deviation is approximately 8.65.
;
Create frequency distribution table: Count data points in intervals 21-25, 26-30, 31-35, 36-40, 41-45, 46-50.
Calculate the mode: Identify the interval with the highest frequency.
Calculate the mean: Mean = 50 ∑ i = 1 50 x i = 34.7 .
Calculate the median: Find the average of the 25th and 26th sorted values: Median = 2 33 + 34 = 33.5 .
Calculate the variance: Variance = 50 ∑ i = 1 50 ( x i − Mean ) 2 = 74.77 .
Calculate the standard deviation: Standard Deviation = Variance = 74.77 = 8.65 .
Mean = 34.7 , Median = 33.5 , Variance = 74.77 , Standard Deviation = 8.65
Explanation
Analyze the problem We are given a set of 50 numbers and asked to perform several statistical analyses: 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: 8
26-30: 10
31-35: 9
36-40: 9
41-45: 6
46-50: 8
Calculate the mode Now, let's calculate the mode. The mode is the interval with the highest frequency, which is 26-30 with a frequency of 10.
Calculate the mean Next, we calculate the mean. The mean is the sum of all data points divided by the number of data points (50). Mean = 50 ∑ i = 1 50 x i = 34.7 So, the mean is 34.7.
Calculate the median To calculate the median, we first sort the data in ascending order. Since there are 50 data points, the median is the average of the 25th and 26th values. The sorted data is:
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 value is 33 and the 26th value is 34. Therefore, the median is: Median = 2 33 + 34 = 33.5
Calculate the variance Now, let's calculate the variance. The variance is the average of the squared differences from the mean. Variance = 50 ∑ i = 1 50 ( x i − Mean ) 2 = 74.77 So, the variance is 74.77.
Calculate the standard deviation Finally, we calculate the standard deviation. The standard deviation is the square root of the variance. Standard Deviation = Variance = 74.77 = 8.646964785403025 So, the standard deviation is approximately 8.65.
Final Answer In summary:
Frequency distribution table:
21-25: 8
26-30: 10
31-35: 9
36-40: 9
41-45: 6
46-50: 8
Mode: 26-30
Mean: 34.7
Median: 33.5
Variance: 74.77
Standard deviation: 8.65
Examples
Understanding data distribution is crucial in many real-world scenarios. For example, in education, analyzing student test scores using frequency distribution, mean, median, variance, and standard deviation helps teachers understand the overall performance of the class. The mode indicates the most common score range, the mean provides the average score, the median represents the middle score, the variance measures the spread of the scores, and the standard deviation quantifies the typical deviation from the mean. This analysis helps teachers identify areas where students excel or struggle, allowing them to tailor their teaching methods accordingly and provide targeted support to improve student outcomes.
