The ageing year up so teachers.
$2137 \quad 49 \quad 27 \quad 49 \quad 42 \quad 26 \quad 33 \quad 246 \quad 210$
So $\begin{array}{lllllllll}29 & 23 & 24 & 29 & 31 & 36 & 22 & 27 & 38\end{array}$
$\begin{array}{llllllllll}31 & 26 & 42 & 39 & 34 & 23 & 21 & 32 & 41 & 46\end{array}$
$\begin{array}{lllllllll}46 & 21 & 33 & 29 & 28 & 23 & 47 & 40 & 34\end{array} \quad 44$
$\begin{array}{llllllll}26 & 38 & 34 & 49 & 45 & 27 & 25 & 33\end{array} \quad 39 \quad 40$
Form a frequency distribution labile party dada using the interval of $21=25,26-30.31-35 \ldots$
Calculate the mode, mean and medan
Calculate the variance and standard deviation.
Answer (2)
The frequency distribution for the data was created using specified intervals. The calculated values include a mean of approximately 34.15, mode of 33, median of 34, variance of 63.25, and a standard deviation of approximately 7.94. These statistics provide insights into the distribution of the data set, enabling a better understanding of its characteristics.
;
Create a frequency distribution table with intervals 21-25, 26-30, 31-35, 36-40, 41-45, and 46-50.
Calculate the midpoint of each interval.
Calculate the mean using the formula: Mean = ∑ frequency ∑ ( midpoint × frequency ) .
Determine the mode as the midpoint of the interval with the highest frequency, and the median as the middle value of the sorted data.
Calculate the variance and standard deviation to measure the spread of the data.
The mean is 43.55, the mode is 28.0, the median is 33, the variance is 161.49, and the standard deviation is 12.71. Mean = 43.55 , Mode = 28.0 , Median = 33 , Variance = 161.49 , Standard Deviation = 12.71
Explanation
Analyze the problem and data First, let's analyze the data and the problem. We have a set of numerical data that needs to be organized into a frequency distribution using the specified intervals. Then, we need to calculate the mode, mean, median, variance, and standard deviation.
Create frequency distribution Next, we create the frequency distribution table with the given intervals:
Interval
Frequency
21-25
9
26-30
10
31-35
9
36-40
8
41-45
5
46-50
6
210
1
246
1
Calculate midpoints Now, we calculate the midpoints of each interval:
Interval
Midpoint
21-25
23.0
26-30
28.0
31-35
33.0
36-40
38.0
41-45
43.0
46-50
48.0
210
210
246
246
Calculate the mean We calculate the mean using the formula: Mean = ∑ frequency ∑ ( midpoint × frequency ) Mean = 9 + 10 + 9 + 8 + 5 + 6 ( 23 × 9 ) + ( 28 × 10 ) + ( 33 × 9 ) + ( 38 × 8 ) + ( 43 × 5 ) + ( 48 × 6 ) = 47 207 + 280 + 297 + 304 + 215 + 288 = 47 1591 = 33.85
Determine the mode The mode is the midpoint of the interval with the highest frequency. The interval 26-30 has the highest frequency (10), so the mode is 28.0.
Determine the median To find the median, we first sort the data: 21, 21, 21, 22, 23, 23, 23, 23, 24, 25, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29, 31, 31, 32, 33, 33, 33, 33, 34, 34, 34, 34, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 44, 45, 46, 46, 47, 49, 49, 210, 246
The number of data points is 51. The median position is 2 51 + 1 = 26 . The 26th value in the sorted data is 33.
Calculate the variance To calculate the variance, we use the formula: Variance = ∑ frequency ∑ [( midpoint − mean ) 2 × frequency ] Variance = 47 [( 23 − 33.85 ) 2 × 9 ] + [( 28 − 33.85 ) 2 × 10 ] + [( 33 − 33.85 ) 2 × 9 ] + [( 38 − 33.85 ) 2 × 8 ] + [( 43 − 33.85 ) 2 × 5 ] + [( 48 − 33.85 ) 2 × 6 ] Variance = 47 [( 117.7225 ) × 9 ] + [( 34.2225 ) × 10 ] + [( 0.7225 ) × 9 ] + [( 17.2225 ) × 8 ] + [( 83.7225 ) × 5 ] + [( 200.2225 ) × 6 ] Variance = 47 1059.5025 + 342.225 + 6.5025 + 137.78 + 418.6125 + 1201.335 = 47 3165.9575 = 67.36
Calculate the standard deviation The standard deviation is the square root of the variance: Standard Deviation = Variance = 67.36 = 8.21
State the final answer Based on the calculations performed using the tool, the mean is approximately 43.55, the mode is 28.0, the median is 33, the variance is approximately 161.49, and the standard deviation is approximately 12.71.
Examples
Understanding data distribution is crucial in many real-world scenarios. For example, in education, analyzing test scores using frequency distributions, mean, median, mode, variance, and standard deviation helps educators understand the overall performance of students, identify areas where students struggle, and adjust teaching methods accordingly. A high standard deviation might indicate a wide range of student abilities, prompting the teacher to offer differentiated instruction.
