Create a histogram with five intervals from this data set:
21, 37, 56, 87, 32, 45, 77, 71, 65, 67, 21, 3
Step 1: Find the minimum and maximum values.
Minimum:
Answer (2)
Find the minimum (3) and maximum (87) values of the data set.
Calculate the range: 87 − 3 = 84 .
Determine the interval width: 5 84 = 16.8 , rounded up to 17.
Count the frequency of each interval: 3-20 (1), 20-37 (4), 37-54 (1), 54-71 (4), 71-88 (2).
The frequency of the minimum value 3 is 1 .
Explanation
Understand the problem and provided data We are given the data set: 21, 37, 56, 87, 32, 45, 77, 71, 65, 67, 21, 3. Our goal is to create a histogram with 5 intervals from this data set.
Find min and max values First, we need to find the minimum and maximum values in the data set. The minimum value is 3, and the maximum value is 87.
Calculate the range Next, we calculate the range of the data set by subtracting the minimum value from the maximum value: r an g e = ma x − min = 87 − 3 = 84
Determine the interval width Now, we determine the interval width by dividing the range by the number of intervals (5). Since we want whole number interval widths, we will round up to the nearest whole number: in t er v a l _ w i d t h = 5 r an g e = 5 84 = 16.8 Rounding up, we get an interval width of 17.
Define the interval boundaries We define the interval boundaries based on the minimum value and the interval width. The intervals are:
3 - (3 + 17) = 3 - 20
20 - (20 + 17) = 20 - 37
37 - (37 + 17) = 37 - 54
54 - (54 + 17) = 54 - 71
71 - (71 + 17) = 71 - 88
Count the frequency of each interval Now, we count the number of data points that fall into each interval (frequency):
3 - 20: 3 (frequency: 1)
20 - 37: 21, 37, 32, 21 (frequency: 4)
37 - 54: 45 (frequency: 1)
54 - 71: 56, 65, 67, 71 (frequency: 4)
71 - 88: 77, 87 (frequency: 2)
Present the histogram Finally, we present the histogram with the intervals and their corresponding frequencies:
3 - 20: 1
20 - 37: 4
37 - 54: 1
54 - 71: 4
71 - 88: 2
Find the frequency of the minimum value The frequency of the minimum value, 3, in the dataset is 1.
Examples
Histograms are useful for visualizing data distributions, such as the distribution of student test scores in a class. By grouping the scores into intervals, we can quickly see how many students fall into each score range, providing insights into the overall performance of the class. This helps teachers identify areas where students may need additional support and tailor their instruction accordingly. For example, if a histogram of test scores shows that a large number of students scored between 60 and 70, the teacher might focus on reviewing the concepts covered in that range.
To create a histogram from the given data set, we identified the minimum value as 3 and the maximum as 87. We calculated the range as 84, determined an interval width of 17, and defined five intervals, counting the frequency of data points in each. The final data showed the frequencies for each interval clearly, allowing us to visualize the distribution of the data set.
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