Question

find the mean, median, mode, range, and iqr of the data set below.
82 93 77 79 90 82 85 85 95 73
79 83 89
mean choose
median choose
mode choose
range choose
iqr (inter quartile range) choose

Explanation:

Step1: Organize the data set

First, we list out the data set and sort it in ascending order. The data set is: 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95 (Wait, no, wait. Wait the original data is 82, 93, 77, 79, 90, 82, 85, 85, 95, 73, 79, 83, 89. Let's count the number of data points. Let's list all: 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95? Wait no, wait the original data: 82, 93, 77, 79, 90, 82, 85, 85, 95, 73, 79, 83, 89. Wait, let's count: 82 (1), 93 (2), 77 (3), 79 (4), 90 (5), 82 (6), 85 (7), 85 (8), 95 (9), 73 (10), 79 (11), 83 (12), 89 (13). Wait, that's 13 data points? Wait no, the first line: 82, 93, 77, 79, 90, 82, 85, 85, 95, 73 (that's 10), then the second line: 79, 83, 89 (that's 3), so total 13 data points. Let's sort them: 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95. Wait, no, 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95? Wait, 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95. Wait, that's 13 numbers. Let's confirm: 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95. Yes, 13 numbers.

Step2: Calculate the Mean

The mean is the sum of all data points divided by the number of data points. Let's calculate the sum:

73 + 77 + 79 + 79 + 82 + 82 + 83 + 85 + 85 + 89 + 90 + 93 + 95.

Let's compute step by step:

73 + 77 = 150

150 + 79 = 229

229 + 79 = 308

308 + 82 = 390

390 + 82 = 472

472 + 83 = 555

555 + 85 = 640

640 + 85 = 725

725 + 89 = 814

814 + 90 = 904

904 + 93 = 997

997 + 95 = 1092

Wait, that can't be right. Wait, maybe I miscounted the data points. Wait the original data: first line: 82, 93, 77, 79, 90, 82, 85, 85, 95, 73 (10 numbers), second line: 79, 83, 89 (3 numbers). So total 13 numbers. Wait, but let's check again. Wait 82, 93, 77, 79, 90, 82, 85, 85, 95, 73: that's 10. Then 79, 83, 89: 3. So 13. Wait, but when I sum them: 73 + 77 = 150; 150 + 79 = 229; 229 + 79 = 308; 308 + 82 = 390; 390 + 82 = 472; 472 + 83 = 555; 555 + 85 = 640; 640 + 85 = 725; 725 + 89 = 814; 814 + 90 = 904; 904 + 93 = 997; 997 + 95 = 1092. Wait, but that's 13 numbers? Wait no, wait 82 (1), 93 (2), 77 (3), 79 (4), 90 (5), 82 (6), 85 (7), 85 (8), 95 (9), 73 (10), 79 (11), 83 (12), 89 (13). Yes, 13. So sum is 1092? Wait, let's check with another approach. Let's list all numbers:

73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95.

Wait, 73 + 95 = 168; 77 + 93 = 170; 79 + 90 = 169; 79 + 89 = 168; 82 + 85 = 167; 82 + 85 = 167; 83. Wait, that's 6 pairs and one single. Wait 168 + 170 = 338; 338 + 169 = 507; 507 + 168 = 675; 675 + 167 = 842; 842 + 167 = 1009; 1009 + 83 = 1092. Yes, sum is 1092. Then mean is 1092 divided by 13. Let's compute that: 13 * 84 = 1092. Oh! So mean is 84.

Step3: Find the Median

Median is the middle value when data is sorted. Since there are 13 data points (odd number), the median is the (13 + 1)/2 = 7th term. Let's check the sorted data:

1: 73

2: 77

3: 79

4: 79

5: 82

6: 82

7: 83

Wait, wait, no. Wait the sorted data: 73, 77, 79, 79, 82, 82, 83, 85, 85, 89, 90, 93, 95. Wait, the 7th term is 83? Wait no, wait 1:73, 2:77, 3:79, 4:79, 5:82, 6:82, 7:83, 8:85, 9:85, 10:89, 11:90, 12:93, 13:95. Yes, the 7th term is 83. So median is 83.

Step4: Find the Mode

Mode is the most frequently occurring value. Let's count the frequency of each number:

73: 1

77: 1

79: 2

82: 2

83: 1

85: 2

89: 1

90: 1

93: 1

95: 1

Wait, 79, 82, 85 each occur 2 times. Wait, but wait the original data: 79 appears twice (positions 4 and 11), 82 appears twice (positions 5 and 6), 85 appears twice (positions 7 and 8). Wait, but wait, maybe I made a mistake. Wait the original data: 82, 93, 77, 79, 90, 82, 85, 85, 95, 73, 79, 83, 89. So 82: two times (first and sixth), 79: two times (fourth and eleventh), 85: two times (seventh and eighth). So mode is 79, 82, 85? Wait, but maybe the problem considers multiple modes, but let's check again. Wait, maybe I miscounted. Wait 82: 82 (1), 82 (6) – two times. 79: 79 (4), 79 (11) – two times. 85: 85 (7), 85 (8) – two times. So mode is 79, 82, 85. But maybe the problem expects the mode as the most frequent, but since all have same frequency, but maybe I made a mistake. Wait, wait, no, wait the original data: let's list all:

82, 93, 77, 79, 90, 82, 85, 85, 95, 73, 79, 83, 89.

So numbers: 82 (2), 93 (1), 77 (1), 79 (2), 90 (1), 85 (2), 95 (1), 73 (1), 83 (1), 89 (1). So yes, 82, 79, 85 each occur twice. So mode is 79, 82, 85. But maybe the problem has a typo, or maybe I miscounted. Wait, wait, maybe the data set is 14 numbers? Wait, let's check again. The first line: 82, 93, 77, 79, 90, 82, 85, 85, 95, 73 (10 numbers). Second line: 79, 83, 89 (3 numbers). Wait, that's 13. Wait, maybe the user made a typo, but according to the given data, it's 13. Wait, but let's proceed.

Step5: Calculate the Range

Range is maximum minus minimum. Maximum is 95, minimum is 73. So range is 95 - 73 = 22.

Step6: Calculate the IQR

IQR is Q3 - Q1. First, we need to find Q1 and Q3. For a data set with n = 13, the median is the 7th term (83). Then Q1 is the median of the lower half (data points 1 to 6), and Q3 is the median of the upper half (data points 8 to 13). Wait, no: when n is odd, we exclude the median when finding Q1 and Q3. So lower half: data points 1 to 6 (73, 77, 79, 79, 82, 82). The median of this lower half (n = 6, even) is the average of the 3rd and 4th terms. 3rd term: 79, 4th term: 79. So Q1 = (79 + 79)/2 = 79. Upper half: data points 8 to 13 (85, 85, 89, 90, 93, 95). The median of this upper half (n = 6, even) is the average of the 3rd and 4th terms. 3rd term: 89, 4th term: 90. So Q3 = (89 + 90)/2 = 89.5. Then IQR = Q3 - Q1 = 89.5 - 79 = 10.5. Wait, but let's check again. Wait, the sorted data:

1: 73

2: 77

3: 79

4: 79

5: 82

6: 82

7: 83 (median)

8: 85

9: 85

10: 89

11: 90

12: 93

13: 95

Lower half: positions 1-6: [73, 77, 79, 79, 82, 82]. Median of lower half: (79 + 79)/2 = 79 (Q1). Upper half: positions 8-13: [85, 85, 89, 90, 93, 95]. Median of upper half: (89 + 90)/2 = 89.5 (Q3). So IQR = 89.5 - 79 = 10.5.

Wait, but let's confirm the steps again.

Answer:

Mean: 84
Median: 83
Mode: 79, 82, 85 (or each with frequency 2)
Range: 22
IQR: 10.5

But let's check the mean again. Sum is 1092, divided by 13: 1092 / 13 = 84. Correct. Median: 7th term is 83. Correct. Mode: 79, 82, 85 each occur twice. Range: 95 - 73 = 22. Correct. IQR: Q3 - Q1 = 89.5 - 79 = 10.5. Correct.