Question

for questions 1 through 3, find the mean, median, mode, and range for each set of data.

1. 10, 71, 83, 97, 38, 55, 74, 5, 59, 97
2. 769, 818, 826, 786, 751, 803, 769, 759, 201
3. 159, 109, 142, 64, 75, 70, 88, 130, 131, 72, 86

Explanation:

Step1: Organize the data

First, we need to sort the data set: \(5, 10, 11, 14, 38, 55, 59, 74, 83, 91, 91\) (Wait, wait, the original data is 10, 11, 83, 91, 38, 55, 74, 5, 59, 91. Let's count the number of data points: 10 numbers? Wait, 10, 11, 83, 91, 38, 55, 74, 5, 59, 91. So that's 10 numbers? Wait, no, let's list them: 5, 10, 11, 38, 55, 59, 74, 83, 91, 91. Wait, that's 10 numbers? Wait, 5, 10, 11, 38, 55, 59, 74, 83, 91, 91. Yes, 10 numbers.

Step2: Calculate the Mean

Mean is the sum of all data points divided by the number of data points. Let's sum them: \(5 + 10 + 11 + 38 + 55 + 59 + 74 + 83 + 91 + 91\). Let's calculate step by step: 5+10=15; 15+11=26; 26+38=64; 64+55=119; 119+59=178; 178+74=252; 252+83=335; 335+91=426; 426+91=517. Wait, no, wait: 5 + 10 = 15; 15 + 11 = 26; 26 + 38 = 64; 64 + 55 = 119; 119 + 59 = 178; 178 + 74 = 252; 252 + 83 = 335; 335 + 91 = 426; 426 + 91 = 517. Wait, but the number of data points is 10? Wait, original data: 10, 11, 83, 91, 38, 55, 74, 5, 59, 91. So that's 10 numbers. So mean is \( \frac{5 + 10 + 11 + 38 + 55 + 59 + 74 + 83 + 91 + 91}{10} \). Let's recalculate the sum: 5 + 10 = 15; 15 + 11 = 26; 26 + 38 = 64; 64 + 55 = 119; 119 + 59 = 178; 178 + 74 = 252; 252 + 83 = 335; 335 + 91 = 426; 426 + 91 = 517. Wait, 517 divided by 10 is 51.7? Wait, that seems low, but let's check the data: 5, 10, 11, 38, 55, 59, 74, 83, 91, 91. Yes, the small numbers are 5,10,11,38, then 55,59,74,83,91,91. So sum is 5+10=15, +11=26, +38=64, +55=119, +59=178, +74=252, +83=335, +91=426, +91=517. So mean is 517/10 = 51.7.

Step3: Calculate the Median

Median is the middle value when the data is sorted. Since there are 10 data points (even number), the median is the average of the 5th and 6th values. Sorted data: 5, 10, 11, 38, 55, 59, 74, 83, 91, 91. The 5th value is 55, the 6th is 59. So median is (55 + 59)/2 = 114/2 = 57.

Step4: Calculate the Mode

Mode is the most frequently occurring value. In the data set, 91 appears twice, others appear once. So mode is 91.

Step5: Calculate the Range

Range is the maximum value minus the minimum value. Maximum is 91, minimum is 5. So range is 91 - 5 = 86.

Wait, but let's check the original data again: the problem says "10, 11, 83, 91, 38, 55, 74, 5, 59, 91". So that's 10 numbers. So the calculations above are for question 1. Let's confirm:

Sum: 5 + 10 + 11 + 38 + 55 + 59 + 74 + 83 + 91 + 91. Let's add again: 5+10=15; 15+11=26; 26+38=64; 64+55=119; 119+59=178; 178+74=252; 252+83=335; 335+91=426; 426+91=517. Correct. So mean is 517/10 = 51.7.

Median: sorted data: 5,10,11,38,55,59,74,83,91,91. 5th term: 55, 6th term: 59. (55+59)/2=57. Correct.

Mode: 91 appears twice, others once. So mode is 91.

Range: 91 - 5 = 86.

Now, let's do question 2: Data set: 769, 818, 826, 786, 751, 803, 769, 759, 201. Wait, wait, the original data is "769, 818, 826, 786, 751, 803, 769, 759, 201". Wait, that's 9 numbers? Wait, let's count: 769, 818, 826, 786, 751, 803, 769, 759, 201. So 9 numbers.

Step1: Sort the data for question 2

Let's sort them: 201, 751, 759, 769, 769, 786, 803, 818, 826.

Step2: Calculate the Mean for question 2

Sum all data points: 201 + 751 + 759 + 769 + 769 + 786 + 803 + 818 + 826. Let's calculate step by step:

201 + 751 = 952;

952 + 759 = 1711;

1711 + 769 = 2480;

2480 + 769 = 3249;

3249 + 786 = 4035;

4035 + 803 = 4838;

4838 + 818 = 5656;

5656 + 826 = 6482.

Number of data points: 9. So mean is 6482 / 9 ≈ 720.22 (rounded to two decimal places).

Step3: Calculate the Median for question 2

Since there are 9 data points (odd), the median is the 5th value. Sorted data: 201, 751, 759, 769, 769, 786, 803, 818, 826. The 5th value is 769. So median is 769.

Step4: Calculate the Mode for question 2

Mode is the most frequent value. 769 appears twice, others once. So mode is 769.

Step5: Calculate the Range for question 2

Maximum value is 826, minimum is 201. So range is 826 - 201 = 625.

Now, question 3: Data set: 159, 109, 142, 64, 15, 70, 88, 130, 131, 72, 86. Let's count: 11 numbers.

Step1: Sort the data for question 3

Sort them: 15, 64, 70, 72, 86, 88, 109, 130, 131, 142, 159.

Step2: Calculate the Mean for question 3

Sum all data points: 15 + 64 + 70 + 72 + 86 + 88 + 109 + 130 + 131 + 142 + 159. Let's calculate:

15 + 64 = 79;

79 + 70 = 149;

149 + 72 = 221;

221 + 86 = 307;

307 + 88 = 395;

395 + 109 = 504;

504 + 130 = 634;

634 + 131 = 765;

765 + 142 = 907;

907 + 159 = 1066.

Number of data points: 11. So mean is 1066 / 11 = 96.909... ≈ 96.91 (rounded to two decimal places).

Step3: Calculate the Median for question 3

Since there are 11 data points (odd), the median is the 6th value. Sorted data: 15, 64, 70, 72, 86, 88, 109, 130, 131, 142, 159. The 6th value is 88. So median is 88.

Step4: Calculate the Mode for question 3

All values appear once, so there is no mode (or we can say all are modes, but typically, if no value repeats, we say no mode).

Step5: Calculate the Range for question 3

Maximum value is 159, minimum is 15. So range is 159 - 15 = 144.

But the problem says "FOR QUESTIONS 1 THROUGH 3, FIND THE MEAN, MEDIAN, MODE, AND RANGE FOR EACH SET OF DATA." So we need to solve each question. Let's present the answers for each:

Question 1:

Data: 10, 11, 83, 91, 38, 55, 74, 5, 59, 91 (sorted: 5, 10, 11, 38, 55, 59, 74, 83, 91, 91)
- Mean: \( \frac{5 + 10 + 11 + 38 + 55 + 59 + 74 + 83 + 91 + 91}{10} = \frac{517}{10} = 51.7 \)
- Median: \( \frac{55 + 59}{2} = 57 \)
- Mode: 91 (appears twice)
- Range: \( 91 - 5 = 86 \)

Question 2:

Data: 769, 818, 826, 786, 751, 803, 769, 759, 201 (sorted: 201, 751, 759, 769, 769, 786, 803, 818, 826)
- Mean: \( \frac{201 + 751 + 759 + 769 + 769 + 786 + 803 + 818 + 826}{9} = \frac{6482}{9} \approx 720.22 \)
- Median: 769 (5th value)
- Mode: 769 (appears twice)
- Range: \( 826 - 201 = 625 \)

Question 3:

Data: 159, 109, 142, 64, 15, 70, 88, 130, 131, 72, 86 (sorted: 15, 64, 70, 72, 86, 88, 109, 130, 131, 142, 159)
- Mean: \( \frac{15 + 64 + 70 + 72 + 86 + 88 + 109 + 130 + 131 + 142 + 159}{11} = \frac{1066}{11} \approx 96.91 \)
- Median: 88 (6th value)
- Mode: No mode (all values unique)
- Range: \( 159 - 15 = 144 \)

But since the user might want the solution for one of them, but the problem says "FOR QUESTIONS 1 THROUGH 3", so we can present each. However, the initial problem might be to solve question 1, 2, or 3. Let's confirm the data for each.

Wait, the user's image shows:

1. 10, 11, 83, 91, 38, 55, 74, 5, 59, 91 (10 numbers)
2. 769, 818, 826, 786, 751, 803, 769, 759, 201 (9 numbers)
3. 159, 109, 142, 64, 15, 70, 88, 130, 131, 72, 86 (11 numbers)

So, depending on which question the user wants, but since the user provided the image, we can solve each. But let's proceed with question 1 as an example.

Answer:

Question 1:

• Mean: \( 51.7 \)
• Median: \( 57 \)
• Mode: \( 91 \)
• Range: \( 86 \)

Question 2:

• Mean: \( \approx 720.22 \)
• Median: \( 769 \)
• Mode: \( 769 \)
• Range: \( 625 \)

Question 3:

• Mean: \( \approx 96.91 \)
• Median: \( 88 \)
• Mode: No mode (all unique)
• Range: \( 144 \)