Math worksheet for calculating mean, median, mode, range, and standard deviation.

To solve the problem, we need to calculate the mean, median, mode, range, and standard deviation for each set of numbers. Let's go through each part step by step.

---

Problem 1:

Numbers: \(239, 230, 500, 120, 230\)

#### Step 1: Mean
The mean is the average of the numbers.
\[
\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Number of values}}
\]
\[
\text{Sum} = 239 + 230 + 500 + 120 + 230 = 1319
\]
\[
\text{Number of values} = 5
\]
\[
\text{Mean} = \frac{1319}{5} = 263.8
\]

#### Step 2: Median
The median is the middle value when the numbers are arranged in ascending order.
\[
\text{Ascending order: } 120, 230, 230, 239, 500
\]
Since there are 5 numbers (odd), the median is the middle number:
\[
\text{Median} = 230
\]

#### Step 3: Mode
The mode is the number that appears most frequently.
\[
\text{Frequency: } 120 (1), 230 (2), 239 (1), 500 (1)
\]
\[
\text{Mode} = 230
\]

#### Step 4: Range
The range is the difference between the largest and smallest numbers.
\[
\text{Largest number} = 500
\]
\[
\text{Smallest number} = 120
\]
\[
\text{Range} = 500 - 120 = 380
\]

#### Step 5: Standard Deviation
The standard deviation measures the spread of the data. The formula is:
\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
\]
where \( \mu \) is the mean, \( x_i \) are the individual values, and \( N \) is the number of values.

1. Calculate the mean (\( \mu \)): Already calculated as 263.8.
2. Calculate the squared differences from the mean:
\[
(239 - 263.8)^2 = (-24.8)^2 = 615.04
\]
\[
(230 - 263.8)^2 = (-33.8)^2 = 1142.44
\]
\[
(500 - 263.8)^2 = (236.2)^2 = 55782.44
\]
\[
(120 - 263.8)^2 = (-143.8)^2 = 20678.44
\]
\[
(230 - 263.8)^2 = (-33.8)^2 = 1142.44
\]
3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 = 615.04 + 1142.44 + 55782.44 + 20678.44 + 1142.44 = 79360.8
\]
4. Divide by the number of values (\( N = 5 \)):
\[
\frac{\sum (x_i - \mu)^2}{N} = \frac{79360.8}{5} = 15872.16
\]
5. Take the square root:
\[
\sigma = \sqrt{15872.16} \approx 126.0
\]

#### Final Answers for Problem 1:
\[
\boxed{263.8, 230, 230, 380, 126.0}
\]

---

Problem 2:

Numbers: \(32, 54, 87, 12, 30, 70, 13, 40, 80, 72\)

#### Step 1: Mean
\[
\text{Sum} = 32 + 54 + 87 + 12 + 30 + 70 + 13 + 40 + 80 + 72 = 490
\]
\[
\text{Number of values} = 10
\]
\[
\text{Mean} = \frac{490}{10} = 49
\]

#### Step 2: Median
Arrange the numbers in ascending order:
\[
12, 13, 30, 32, 40, 54, 70, 72, 80, 87
\]
Since there are 10 numbers (even), the median is the average of the 5th and 6th numbers:
\[
\text{Median} = \frac{40 + 54}{2} = \frac{94}{2} = 47
\]

#### Step 3: Mode
Check the frequency of each number:
\[
\text{All numbers appear once.}
\]
\[
\text{Mode} = \text{No mode (all unique)}
\]

#### Step 4: Range
\[
\text{Largest number} = 87
\]
\[
\text{Smallest number} = 12
\]
\[
\text{Range} = 87 - 12 = 75
\]

#### Step 5: Standard Deviation
1. Calculate the mean (\( \mu \)): Already calculated as 49.
2. Calculate the squared differences from the mean:
\[
(32 - 49)^2 = (-17)^2 = 289
\]
\[
(54 - 49)^2 = (5)^2 = 25
\]
\[
(87 - 49)^2 = (38)^2 = 1444
\]
\[
(12 - 49)^2 = (-37)^2 = 1369
\]
\[
(30 - 49)^2 = (-19)^2 = 361
\]
\[
(70 - 49)^2 = (21)^2 = 441
\]
\[
(13 - 49)^2 = (-36)^2 = 1296
\]
\[
(40 - 49)^2 = (-9)^2 = 81
\]
\[
(80 - 49)^2 = (31)^2 = 961
\]
\[
(72 - 49)^2 = (23)^2 = 529
\]
3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 = 289 + 25 + 1444 + 1369 + 361 + 441 + 1296 + 81 + 961 + 529 = 6396
\]
4. Divide by the number of values (\( N = 10 \)):
\[
\frac{\sum (x_i - \mu)^2}{N} = \frac{6396}{10} = 639.6
\]
5. Take the square root:
\[
\sigma = \sqrt{639.6} \approx 25.3
\]

#### Final Answers for Problem 2:
\[
\boxed{49, 47, \text{No mode}, 75, 25.3}
\]

---

Problem 3:

Numbers: \(1.4, 2.1, 0.7, 0.4, 0.6, 0.8, 0.9\)

#### Step 1: Mean
\[
\text{Sum} = 1.4 + 2.1 + 0.7 + 0.4 + 0.6 + 0.8 + 0.9 = 6.9
\]
\[
\text{Number of values} = 7
\]
\[
\text{Mean} = \frac{6.9}{7} \approx 0.986
\]

#### Step 2: Median
Arrange the numbers in ascending order:
\[
0.4, 0.6, 0.7, 0.8, 0.9, 1.4, 2.1
\]
Since there are 7 numbers (odd), the median is the middle number:
\[
\text{Median} = 0.8
\]

#### Step 3: Mode
Check the frequency of each number:
\[
\text{All numbers appear once.}
\]
\[
\text{Mode} = \text{No mode (all unique)}
\]

#### Step 4: Range
\[
\text{Largest number} = 2.1
\]
\[
\text{Smallest number} = 0.4
\]
\[
\text{Range} = 2.1 - 0.4 = 1.7
\]

#### Step 5: Standard Deviation
1. Calculate the mean (\( \mu \)): Already calculated as 0.986.
2. Calculate the squared differences from the mean:
\[
(1.4 - 0.986)^2 = (0.414)^2 \approx 0.171
\]
\[
(2.1 - 0.986)^2 = (1.114)^2 \approx 1.241
\]
\[
(0.7 - 0.986)^2 = (-0.286)^2 \approx 0.082
\]
\[
(0.4 - 0.986)^2 = (-0.586)^2 \approx 0.343
\]
\[
(0.6 - 0.986)^2 = (-0.386)^2 \approx 0.149
\]
\[
(0.8 - 0.986)^2 = (-0.186)^2 \approx 0.035
\]
\[
(0.9 - 0.986)^2 = (-0.086)^2 \approx 0.007
\]
3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 \approx 0.171 + 1.241 + 0.082 + 0.343 + 0.149 + 0.035 + 0.007 = 2.028
\]
4. Divide by the number of values (\( N = 7 \)):
\[
\frac{\sum (x_i - \mu)^2}{N} = \frac{2.028}{7} \approx 0.29
\]
5. Take the square root:
\[
\sigma = \sqrt{0.29} \approx 0.54
\]

#### Final Answers for Problem 3:
\[
\boxed{0.986, 0.8, \text{No mode}, 1.7, 0.54}
\]

---

Problem 4:

Numbers: \(7.4, 7.3, 7.5, 7.4, 7.8, 7.3, 7.6, 7.4\)

#### Step 1: Mean
\[
\text{Sum} = 7.4 + 7.3 + 7.5 + 7.4 + 7.8 + 7.3 + 7.6 + 7.4 = 59.7
\]
\[
\text{Number of values} = 8
\]
\[
\text{Mean} = \frac{59.7}{8} = 7.4625
\]

#### Step 2: Median
Arrange the numbers in ascending order:
\[
7.3, 7.3, 7.4, 7.4, 7.4, 7.5, 7.6, 7.8
\]
Since there are 8 numbers (even), the median is the average of the 4th and 5th numbers:
\[
\text{Median} = \frac{7.4 + 7.4}{2} = 7.4
\]

#### Step 3: Mode
Check the frequency of each number:
\[
7.3 (2), 7.4 (3), 7.5 (1), 7.6 (1), 7.8 (1)
\]
\[
\text{Mode} = 7.4
\]

#### Step 4: Range
\[
\text{Largest number} = 7.8
\]
\[
\text{Smallest number} = 7.3
\]
\[
\text{Range} = 7.8 - 7.3 = 0.5
\]

#### Step 5: Standard Deviation
1. Calculate the mean (\( \mu \)): Already calculated as 7.4625.
2. Calculate the squared differences from the mean:
\[
(7.4 - 7.4625)^2 = (-0.0625)^2 \approx 0.0039
\]
\[
(7.3 - 7.4625)^2 = (-0.1625)^2 \approx 0.0264
\]
\[
(7.5 - 7.4625)^2 = (0.0375)^2 \approx 0.0014
\]
\[
(7.4 - 7.4625)^2 = (-0.0625)^2 \approx 0.0039
\]
\[
(7.8 - 7.4625)^2 = (0.3375)^2 \approx 0.1139
\]
\[
(7.3 - 7.4625)^2 = (-0.1625)^2 \approx 0.0264
\]
\[
(7.6 - 7.4625)^2 = (0.1375)^2 \approx 0.0189
\]
\[
(7.4 - 7.4625)^2 = (-0.0625)^2 \approx 0.0039
\]
3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 \approx 0.0039 + 0.0264 + 0.0014 + 0.0039 + 0.1139 + 0.0264 + 0.0189 + 0.0039 = 0.204
\]
4. Divide by the number of values (\( N = 8 \)):
\[
\frac{\sum (x_i - \mu)^2}{N} = \frac{0.204}{8} \approx 0.0255
\]
5. Take the square root:
\[
\sigma = \sqrt{0.0255} \approx 0.16
\]

#### Final Answers for Problem 4:
\[
\boxed{7.4625, 7.4, 7.4, 0.5, 0.16}
\]

---

Problem 5:

Numbers: \(54, 69, 81, 100, 105, 108, 129\)

#### Step 1: Mean
\[
\text{Sum} = 54 + 69 + 81 + 100 + 105 + 108 + 129 = 646
\]
\[
\text{Number of values} = 7
\]
\[
\text{Mean} = \frac{646}{7} \approx 92.29
\]

#### Step 2: Median
Arrange the numbers in ascending order:
\[
54, 69, 81, 100, 105, 108, 129
\]
Since there are 7 numbers (odd), the median is the middle number:
\[
\text{Median} = 100
\]

#### Step 3: Mode
Check the frequency of each number:
\[
\text{All numbers appear once.}
\]
\[
\text{Mode} = \text{No mode (all unique)}
\]

#### Step 4: Range
\[
\text{Largest number} = 129
\]
\[
\text{Smallest number} = 54
\]
\[
\text{Range} = 129 - 54 = 75
\]

#### Step 5: Standard Deviation
1. Calculate the mean (\( \mu \)): Already calculated as 92.29.
2. Calculate the squared differences from the mean:
\[
(54 - 92.29)^2 = (-38.29)^2 \approx 1466.52
\]
\[
(69 - 92.29)^2 = (-23.29)^2 \approx 542.42
\]
\[
(81 - 92.29)^2 = (-11.29)^2 \approx 127.46
\]
\[
(100 - 92.29)^2 = (7.71)^2 \approx 59.44
\]
\[
(105 - 92.29)^2 = (12.71)^2 \approx 161.56
\]
\[
(108 - 92.29)^2 = (15.71)^2 \approx 246.80
\]
\[
(129 - 92.29)^2 = (36.71)^2 \approx 1347.82
\]
3. Sum the squared differences:
\[
\sum (x_i - \mu)^2 \approx 1466.52 + 542.42 + 127.46 + 59.44 + 161.56 + 246.80 + 1347.82 = 3952.02
\]
4. Divide by the number of values (\( N = 7 \)):
\[
\frac{\sum (x_i - \mu)^2}{N} = \frac{3952.02}{7} \approx 564.57
\]
5. Take the square root:
\[
\sigma = \sqrt{564.57} \approx 23.76
\]

#### Final Answers for Problem 5:
\[
\boxed{92.29, 100, \text{No mode}, 75, 23.76}
\]

---

Final Summary of All Problems:

1. \( \boxed{263.8, 230, 230, 380, 126.0} \)
2. \( \boxed{49, 47, \text{No mode}, 75, 25.3} \)
3. \( \boxed{0.986, 0.8, \text{No mode}, 1.7, 0.54} \)
4. \( \boxed{7.4625, 7.4, 7.4, 0.5, 0.16} \)
5. \( \boxed{92.29, 100, \text{No mode}, 75, 23.76} \)