Computational Programming. Part 7

Август 19, 2022

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Computational Programming. Part 7

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2. Finding the Mean The mean is a common and intuitive way to summarize a set of
3. >>> shortlist = [1, 2, 3] >>> sum(shortlist) 6 We can use the len() function to
4. When we use the len() function on the list, it returns 3 because there are three
5. ''' Calculating the mean ''' def calculate_mean(numbers): u s = sum(numbers) v N = len(numbers) #
6. First, we define a function, calculate_mean(), that accepts the argument numbers, which is a list of
7. The calculate_mean() function will calculate the sum and length of any list, so we can reuse
8. Finding the Median The median of a collection of numbers is another kind of average. To
9. Now assume—just for this example—that we have another donation total for the 13th day so that
10. >>> samplelist = [4, 1, 3] >>> samplelist.sort() >>> samplelist [1, 3, 4]
11. Now we can write our next program, which finds the median of a list of numbers:
12. ''' Calculating the median ''' def calculate_median(numbers): u N = len(numbers) v numbers.sort() # Find the
13. The overall structure of the program is similar to that of the earlier program that calculates
14. We cannot use a floating point number as an index in a list, so we use
15. Finding the Most Common Elements Finding the most common number in a data set can be
16. Here, we start off with a list of five numbers and import Counter from the collections
17. >>> c.most_common(1) [(4, 2)] If you call the method again with 2 as an argument, you’ll
18. The most_common() method returns both the numbers and the number of times they occur. What if
19. Finding the Mode We’re ready to write a program that finds the mode for a list
20. The rest of the program applies the calculate_mode function to the list of test scores we
21. What if you have a set of data where two or more numbers occur the same
22. ''' Calculating the mode when the list of numbers may have multiple modes ''' from collections
23. When you execute the preceding program, you should see the following output: The mode(s) of the
24. Creating a Frequency Table Let’s consider the list of test scores again: 7, 8, 9, 2,
25. Here’s the program: ''' Frequency table for a list of numbers ''' from collections import Counter
26. The function frequency_table() prints the frequency table of the list of numbers passed to it. At
27. Number Frequency 9 5 6 3 1 2 5 2 7 2 8 2 10 2
28. Here, you can see that the numbers are listed in decreasing order of frequency because the
29. ''' Frequency table for a list of numbers Enhanced to display the table sorted by the
30. Here, we store the list returned by the most_common() method in numbers_freq at u, and then
31. Number Frequency 1 2 2 1 4 1 5 2 6 3 7 2 8 2
32. Measuring the Dispersion The next statistical calculations we’ll look at measure the dispersion, which tells us
33. Finding the Range of a Set of Numbers Once again, consider the list of donations during
34. The next program finds the range of the preceding list of donations: ''' Find the range
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Finding the Mean The mean is a common and intuitive way to

summarize a set of numbers. It’s what we might simply call the “average” in everyday use, although as we’ll see, there are other kinds of averages as well. Let’s take a sample set of numbers and calculate the mean. Say there’s a school charity that’s been taking donations over a period of time spanning the last 12 days (we’ll refer to this as period A). In that time, the following 12 numbers represent the total dollar amount of donations received for each day: 100, 60, 70, 900, 100, 200, 500, 500, 503, 600, 1000, and 1200. We can calculate the mean by summing these totals and then dividing the sum by the number of days. In this case, the sum of the numbers is 5733. If we divide this number by 12 (the number of days), we get 477.75, which is the mean donation per day. This number gives us a general idea of how much money was donated on any given day. In a moment, we’ll write a program that calculates and prints the mean for a collection of numbers. As we just saw, to calculate the mean, we’ll need to take the sum of the list of numbers and divide it by the number of items in the list. Let’s look at two Python functions that make both of these operations very easy: sum() and len().
When you use the sum() function on a list of numbers, it adds up all the numbers in the list and returns the result:

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>>> shortlist = [1, 2, 3] >>> sum(shortlist) 6
We can use the

len() function to give us the length of a list:
>>> len(shortlist)
3

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When we use the len() function on the list, it returns

3 because there are three items in shortlist. Now we’re ready to write a program that will calculate the mean of the list of donations

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''' Calculating the mean ''' def calculate_mean(numbers): u s = sum(numbers) v N = len(numbers) # Calculate

the mean w mean = s/N return mean if __name__ == '__main__': x donations = [100, 60, 70, 900, 100, 200, 500, 500, 503, 600, 1000, 1200] y mean = calculate_mean(donations) N = len(donations) z print('Mean donation over the last {0} days is {1}'.format(N, mean))

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First, we define a function, calculate_mean(), that accepts the argument numbers, which

is a list of numbers. At u, we use the sum() function to add up the numbers in the list and create a label, s, to refer to the total. Similarly, at v, we use the len() function to get the length of the list and create a label, N, to refer to it. Then, as you can see at w, we calculate the mean by simply dividing the sum (s) by the number of members (N). At x, we create a list, donations, with the values of the donations listed earlier. We then call the calculate_mean() function, passing this list as an argument at y. Finally, we print the mean that was calculated at z. When you run the program, you should see the following:

Mean donation over the last 12 days is 477.75

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The calculate_mean() function will calculate the sum and length of any list,

so we can reuse it to calculate the mean for other sets of numbers, too. We calculated that the mean donation per day was 477.75. It’s worth noting that the donations during the first few days were much lower than the mean donation we calculated and that the donations during the last couple of days were much higher. The mean gives us one way to summarize the data, but it doesn’t give us a full picture. There are other statistical measurements, however, that can tell us more about the data when compared with the mean.

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Finding the Median The median of a collection of numbers is another

kind of average. To find the median, we sort the numbers in ascending order. If the length of the list of numbers is odd, the number in the middle of the list is the median. If the length of the list of numbers is even, we get the median by taking the mean of the two middle numbers. Let’s find the median of the previous list of donations: 100, 60, 70, 900, 100, 200, 500, 500, 503, 600, 1000, and 1200. After sorting from smallest to largest, the list of numbers becomes 60, 70, 100, 100, 200, 500, 500, 503, 600, 900, 1000, and 1200. We have an even number of items in the list (12), so to get the median, we need to take the mean of the two middle numbers. In this case, the middle numbers are the sixth and the seventh numbers—500 and 500—and the mean of these two numbers is (500 + 500)/2, which comes out to 500. That means the median is 500.

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Now assume—just for this example—that we have another donation total for the

13th day so that the list now looks like this: 100, 60, 70, 900, 100, 200, 500, 500, 503, 600, 1000, 1200, and 800. Once again, we have to sort the list, which becomes 60, 70, 100, 100, 200, 500, 500, 503, 600, 800, 900, 1000, and 1200. There are 13 numbers in this list (an odd number), so the median for this list is simply the middle number. In this case, that’s the seventh number, which is 500. Before we write a program to find the median of a list of numbers, let’s think about how we could automatically calculate the middle elements of a list in either case. If the length of a list (N) is odd, the middle number is the one in position (N + 1)/2. If N is even, the two middle elements are N/2 and (N/2) + 1. For our first example in this section, N = 12, so the two middle elements were the 12/2 (sixth) and 12/2 + 1 (seventh) elements. In the second example, N = 13, so the seventh element, (N + 1)/2, was the middle element. In order to write a function that calculates the median, we’ll also need to sort a list in ascending order. Luckily, the sort() method does just that:

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>>> samplelist = [4, 1, 3] >>> samplelist.sort() >>> samplelist [1, 3, 4]

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Now we can write our next program, which finds the median

of a list of numbers:

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''' Calculating the median ''' def calculate_median(numbers): u N = len(numbers) v numbers.sort() # Find the median if

N % 2 == 0: # if N is even m1 = N/2 m2 = (N/2) + 1 # Convert to integer, match position w m1 = int(m1) - 1 x m2 = int(m2) - 1 y median = (numbers[m1] + numbers[m2])/2 else: z m = (N+1)/2 # Convert to integer, match position m = int(m) - 1 median = numbers[m] return median if __name__ == '__main__': donations = [100, 60, 70, 900, 100, 200, 500, 500, 503, 600, 1000, 1200] median = calculate_median(donations) N = len(donations) print('Median donation over the last {0} days is {1}'.format(N, median))

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The overall structure of the program is similar to that of

the earlier program that calculates the mean. The calculate_median() function accepts a list of numbers and returns the median. At u, we calculate the length of the list and create a label, N, to refer to it. Next, at v, we sort the list using the sort() method. Then, we check to see whether N is even. If so, we find the middle elements, m1 and m2, which are the numbers at positions N/2 and (N/2) + 1 in the sorted list. The next two statements (w and x) adjust m1 and m2 in two ways. First, we use the int() function to convert m1 and m2 into integer form. This is because results of the division operator are always returned as floating point numbers, even when the result is equivalent to an integer. For example: >>> 6/2 3.0

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We cannot use a floating point number as an index in

a list, so we use int() to convert that result to an integer. We also subtract 1 from both m1 and m2 because positions in a list begin with 0 in Python. This means that to get the sixth and seventh numbers from the list, we have to ask for the numbers at index 5 and index 6. At y, we calculate the median by taking the mean of the two numbers in the middle positions. Starting at z, the program finds the median if there’s an odd number of items in the list, once again using int() and subtracting 1 to find the proper index. Finally, the program calculates the median for the list of donations and returns it. When you execute the program, it calculates that the median is 500: Median donation over the last 12 days is 500.0

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Finding the Most Common Elements Finding the most common number in a

data set can be thought of as a subproblem of finding an arbitrary number of most common numbers. For instance, instead of the most common score, what if you wanted to know the five most common scores? The most_common() method of the Counter class allows us to answer such questions easily. Let’s see an example: >>> simplelist = [4, 2, 1, 3, 4] >>> from collections import Counter >>> c = Counter(simplelist) >>> c.most_common() [(4, 2), (1, 1), (2, 1), (3, 1)]

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Here, we start off with a list of five numbers and

import Counter from the collections module. Then, we create a Counter object, using c to refer to the object. We then call the most_common() method, which returns a list ordered by the most common elements. Each member of the list is a tuple. The first element of the first tuple is the number that occurs most frequently, and the second element is the number of times it occurs. The second, third, and fourth tuples contain the other numbers along with the count of the number of times they appear. This result tells us that 4 occurs the most (twice), while the others appear only once. Note that numbers that occur an equal number of times are returned by the most_common() method in an arbitrary order. When you call the most_common() method, you can also provide an argument telling it the number of most common elements you want it to return. For example, if we just wanted to find the most common element, we would call it with the argument 1:

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>>> c.most_common(1) [(4, 2)] If you call the method again with 2 as

an argument, you’ll see this: >>> c.most_common(2) [(4, 2), (1, 1)] Now the result returned by the most_common method is a list with two tuples. The first is the most common element, followed by the second most common. Of course, in this case, there are several elements tied for most common, so the fact that the function returns 1 here (and not 2 or 3) is arbitrary, as noted earlier.

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The most_common() method returns both the numbers and the number of times

they occur. What if we want only the numbers and we don’t care about the number of times they occur? Here’s how we can retrieve that information: >>> mode = c.most_common(1) >>> mode [(4, 2)] >>> mode[0] (4, 2) >>> mode[0][0] 4

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Finding the Mode We’re ready to write a program that finds the

mode for a list of numbers: ''' Calculating the mode ''' from collections import Counter def calculate_mode(numbers): c = Counter(numbers) mode = c.most_common(1) return mode[0][0] if __name__=='__main__': scores = [7, 8, 9, 2, 10, 9, 9, 9, 9, 4, 5, 6, 1, 5, 6, 7, 8, 6, 1, 10] mode = calculate_mode(scores) print('The mode of the list of numbers is: {0}'.format(mode))

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The rest of the program applies the calculate_mode function to the

list of test scores we saw earlier. When you run the program, you should see the following output: The mode of the list of numbers is: 9

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What if you have a set of data where two or

more numbers occur the same maximum number of times? For example, in the list of numbers 5, 5,5, 4, 4, 4, 9, 1, and 3, both 4 and 5 are present three times. In such cases, the list of numbers is said to have multiple modes, and our program should find and print all the modes. The modified program follows:

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''' Calculating the mode when the list of numbers may have multiple modes ''' from

collections import Counter def calculate_mode(numbers): c = Counter(numbers) numbers_freq = c.most_common() max_count = numbers_freq[0][1] modes = [] for num in numbers_freq: if num[1] == max_count: modes.append(num[0]) return modes if __name__ == '__main__': scores = [5, 5, 5, 4, 4, 4, 9, 1, 3] modes = calculate_mode(scores) print('The mode(s) of the list of numbers are:') for mode in modes: print(mode)

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When you execute the preceding program, you should see the following

output: The mode(s) of the list of numbers are: 4 5 What if you wanted to find the number of times every number occurs instead of just the mode? A frequency table, as the name indicates, is a table that shows how many times each number occurs within a collection of numbers.

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Creating a Frequency Table Let’s consider the list of test scores again:

7, 8, 9, 2, 10, 9, 9, 9, 9, 4, 5, 6, 1, 5, 6, 7, 8, 6, 1, and 10. The frequency table for this list is shown in Table 3-1. For each number, we list the number of times it occurs in the second column. Table 3-1: Frequency Table Score Frequency 1 2 2 1 4 1 5 2 6 3 7 2 8 2 9 5 10 2 Note that the sum of the individual frequencies in the second column adds up to the total number of scores (in this case, 20).

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Here’s the program: ''' Frequency table for a list of numbers ''' from collections import

Counter def frequency_table(numbers): u table = Counter(numbers) print('Number\tFrequency') v for number in table.most_common(): print('{0}\t{1}'.format(number[0], number[1])) if __name__=='__main__': scores = [7, 8, 9, 2, 10, 9, 9, 9, 9, 4, 5, 6, 1, 5, 6, 7, 8, 6, 1, 10] frequency_table(scores)

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The function frequency_table() prints the frequency table of the list of

numbers passed to it. At u, we first create a Counter object and create the label table to refer to it. Next, using a for loop v, we go through each of the tuples, printing the first member (the number itself) and the second member (the frequency of the corresponding number). We use \t to print a tab between each value to space the table. When you run the program, you’ll see the following output:

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Number Frequency 9 5 6 3 1 2 5 2 7 2 8 2 10 2 2 1 4 1

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Here, you can see that the numbers are listed in decreasing

order of frequency because the most_common() function returns the numbers in this order. If, instead, you want your program to print the frequency table sorted by value from lowest to highest, as shown in Table 3-1, you’ll have to re-sort the list of tuples. The sort() method is all we need to modify our earlier frequency table program:

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''' Frequency table for a list of numbers Enhanced to display the table

sorted by the numbers ''' from collections import Counter def frequency_table(numbers): table = Counter(numbers) numbers_freq = table.most_common() numbers_freq.sort() print('Number\tFrequency') for number in numbers_freq: print('{0}\t{1}'.format(number[0], number[1])) Describing Data with Statistics 71 if __name__ == '__main__': scores = [7, 8, 9, 2, 10, 9, 9, 9, 9, 4, 5, 6, 1, 5, 6, 7, 8, 6, 1, 10] frequency_table(scores)

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Here, we store the list returned by the most_common() method in numbers_freq

at u, and then we sort it by calling the sort() method v. Finally, we use the for loop to go over the sorted tuples and print each number and its frequency w. Now when you run the program, you’ll see the following table, which is identical to Table 3-1:

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Number Frequency 1 2 2 1 4 1 5 2 6 3 7 2 8 2 9 5 10 2

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Measuring the Dispersion The next statistical calculations we’ll look at measure the

dispersion, which tells us how far away the numbers in a set of data are from the mean of the data set. We’ll learn to calculate three different measurements of dispersion: range, variance, and standard deviation.

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Finding the Range of a Set of Numbers Once again, consider the

list of donations during period A: 100, 60, 70, 900,100, 200, 500, 500, 503, 600, 1000, and 1200. We found that the mean donation per day is 477.75. But just looking at the mean, we have no idea whether all the donations fell into a narrow range—say between 400 and 500—or whether they varied much more than that—say between 60 and 1200, as in this case. For a list of numbers, the range is the difference between the highest number and the lowest number. You could have two groups of numbers with the exact same mean but with vastly different ranges, so knowing the range fills in more information about a set of numbers beyond what we can learn from just looking at the mean, median, and mode.

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The next program finds the range of the preceding list of

donations: ''' Find the range ''' def find_range(numbers): lowest = min(numbers) highest = max(numbers) # Find the range r = highest-lowest return lowest, highest, r if __name__ == '__main__': donations = [100, 60, 70, 900, 100, 200, 500, 500, 503, 600, 1000, 1200] lowest, highest, r = find_range(donations) print('Lowest: {0} Highest: {1} Range: {2}'.format(lowest, highest, r))