Question
Question 2
The monthly cost (K’000) of dinner at Pamodzi Hotel was recorded for 10 tourists and dignitaries who came to pay their last respect the late republican President.
35 56 33 30 31 56 29 27 21 32
Using the information above, calculate
a) Mode, Mean, and Median
b) Variance, Standard Deviation and Coefficient of Variation
The monthly cost (K’000) of dinner at Pamodzi Hotel was recorded for 10 tourists and dignitaries who came to pay their last respect the late republican President.
35 56 33 30 31 56 29 27 21 32
Using the information above, calculate
a) Mode, Mean, and Median
b) Variance, Standard Deviation and Coefficient of Variation
Answers
GPT-4o mini
To calculate the statistics requested for the monthly costs of dinner at Pamodzi Hotel, we start with the given data:
**Data:**
35, 56, 33, 30, 31, 56, 29, 27, 21, 32
### a) Mode, Mean, and Median
**1. Mode:**
The mode is the value that appears most frequently in the data set.
- Looking at the data, the number 56 appears twice while all other values appear once.
- Therefore, the **Mode = 56**.
**2. Mean:**
The mean is calculated by summing all the values and dividing by the number of values.
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{35 + 56 + 33 + 30 + 31 + 56 + 29 + 27 + 21 + 32}{10}
\]
\[
= \frac{359}{10} = 35.9
\]
Thus, the **Mean = 35.9**.
**3. Median:**
To determine the median, we need to sort the data in ascending order:
21, 27, 29, 30, 31, 32, 33, 35, 56, 56
Since there are 10 values (an even number), the median will be the average of the 5th and 6th values.
\[
\text{Median} = \frac{31 + 32}{2} = \frac{63}{2} = 31.5
\]
Thus, the **Median = 31.5**.
### Summary of a:
- Mode: 56
- Mean: 35.9
- Median: 31.5
### b) Variance, Standard Deviation, and Coefficient of Variation
**1. Variance:**
To find variance, we first calculate the mean (already done above: Mean = 35.9), then compute the squared differences from the mean.
\[
\begin{align*}
\text{Values} & : 35, 56, 33, 30, 31, 56, 29, 27, 21, 32 \\
\text{Mean} & : 35.9 \\
\text{Squared Differences} & : \\
(35 - 35.9)^2 & = (−0.9)^2 = 0.81 \\
(56 - 35.9)^2 & = (20.1)^2 = 404.01 \\
(33 - 35.9)^2 & = (−2.9)^2 = 8.41 \\
(30 - 35.9)^2 & = (−5.9)^2 = 34.81 \\
(31 - 35.9)^2 & = (−4.9)^2 = 24.01 \\
(56 - 35.9)^2 & = (20.1)^2 = 404.01 \\
(29 - 35.9)^2 & = (−6.9)^2 = 47.61 \\
(27 - 35.9)^2 & = (−8.9)^2 = 79.21 \\
(21 - 35.9)^2 & = (−14.9)^2 = 222.01 \\
(32 - 35.9)^2 & = (−3.9)^2 = 15.21 \\
\end{align*}
\]
Now, sum of squared differences:
\[
0.81 + 404.01 + 8.41 + 34.81 + 24.01 + 404.01 + 47.61 + 79.21 + 222.01 + 15.21 = 1222.7
\]
Now, calculate variance:
\[
\text{Variance} = \frac{\text{Sum of squared differences}}{N} = \frac{1222.7}{10} = 122.27
\]
Thus, the **Variance = 122.27**.
**2. Standard Deviation:**
Standard deviation is the square root of the variance.
\[
\text{Standard Deviation} = \sqrt{122.27} \approx 11.04
\]
Thus, the **Standard Deviation ≈ 11.04**.
**3. Coefficient of Variation (CV):**
The coefficient of variation is calculated by dividing the standard deviation by the mean and multiplying by 100 to express it as a percentage.
\[
\text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100 = \left( \frac{11.04}{35.9} \right) \times 100 \approx 30.74\%
\]
Thus, the **Coefficient of Variation ≈ 30.74%**.
### Summary of b:
- Variance: 122.27
- Standard Deviation: 11.04
- Coefficient of Variation: 30.74%
**Data:**
35, 56, 33, 30, 31, 56, 29, 27, 21, 32
### a) Mode, Mean, and Median
**1. Mode:**
The mode is the value that appears most frequently in the data set.
- Looking at the data, the number 56 appears twice while all other values appear once.
- Therefore, the **Mode = 56**.
**2. Mean:**
The mean is calculated by summing all the values and dividing by the number of values.
\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{35 + 56 + 33 + 30 + 31 + 56 + 29 + 27 + 21 + 32}{10}
\]
\[
= \frac{359}{10} = 35.9
\]
Thus, the **Mean = 35.9**.
**3. Median:**
To determine the median, we need to sort the data in ascending order:
21, 27, 29, 30, 31, 32, 33, 35, 56, 56
Since there are 10 values (an even number), the median will be the average of the 5th and 6th values.
\[
\text{Median} = \frac{31 + 32}{2} = \frac{63}{2} = 31.5
\]
Thus, the **Median = 31.5**.
### Summary of a:
- Mode: 56
- Mean: 35.9
- Median: 31.5
### b) Variance, Standard Deviation, and Coefficient of Variation
**1. Variance:**
To find variance, we first calculate the mean (already done above: Mean = 35.9), then compute the squared differences from the mean.
\[
\begin{align*}
\text{Values} & : 35, 56, 33, 30, 31, 56, 29, 27, 21, 32 \\
\text{Mean} & : 35.9 \\
\text{Squared Differences} & : \\
(35 - 35.9)^2 & = (−0.9)^2 = 0.81 \\
(56 - 35.9)^2 & = (20.1)^2 = 404.01 \\
(33 - 35.9)^2 & = (−2.9)^2 = 8.41 \\
(30 - 35.9)^2 & = (−5.9)^2 = 34.81 \\
(31 - 35.9)^2 & = (−4.9)^2 = 24.01 \\
(56 - 35.9)^2 & = (20.1)^2 = 404.01 \\
(29 - 35.9)^2 & = (−6.9)^2 = 47.61 \\
(27 - 35.9)^2 & = (−8.9)^2 = 79.21 \\
(21 - 35.9)^2 & = (−14.9)^2 = 222.01 \\
(32 - 35.9)^2 & = (−3.9)^2 = 15.21 \\
\end{align*}
\]
Now, sum of squared differences:
\[
0.81 + 404.01 + 8.41 + 34.81 + 24.01 + 404.01 + 47.61 + 79.21 + 222.01 + 15.21 = 1222.7
\]
Now, calculate variance:
\[
\text{Variance} = \frac{\text{Sum of squared differences}}{N} = \frac{1222.7}{10} = 122.27
\]
Thus, the **Variance = 122.27**.
**2. Standard Deviation:**
Standard deviation is the square root of the variance.
\[
\text{Standard Deviation} = \sqrt{122.27} \approx 11.04
\]
Thus, the **Standard Deviation ≈ 11.04**.
**3. Coefficient of Variation (CV):**
The coefficient of variation is calculated by dividing the standard deviation by the mean and multiplying by 100 to express it as a percentage.
\[
\text{CV} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100 = \left( \frac{11.04}{35.9} \right) \times 100 \approx 30.74\%
\]
Thus, the **Coefficient of Variation ≈ 30.74%**.
### Summary of b:
- Variance: 122.27
- Standard Deviation: 11.04
- Coefficient of Variation: 30.74%