Answer :
To analyze this data set of resistor values, we will calculate the mode, median, mean, and standard deviation step-by-step.
(a) Mode:
The mode is the number that appears most frequently in a data set.
Let's count the frequency of each resistor value:
- 98 Ω: 8 times
- 99 Ω: 18 times
- 100 Ω: 20 times
- 101 Ω: 16 times
- 102 Ω: 12 times
- 103 Ω: 6 times
- 104 Ω: 4 times
- 105 Ω: 1 time
The resistor value that appears most frequently is 100 Ω, appearing 20 times, so the mode is 100 Ω.
(b) Median:
The median is the middle value when a data set is ordered from least to greatest.
- First, list the values in ascending order (partially shown for brevity):
- 95, 95, 96, 96, ..., 100, ..., 105.
- Since there are 100 values, the median will be the average of the 50th and 51st values in the ordered list.
- The 50th value is 100, and the 51st value is also 100.
Therefore, the median is [tex]\frac{100 + 100}{2} = 100[/tex] Ω.
(c) Mean:
The mean is calculated by dividing the sum of all data points by the number of data points.
[tex]\text{Mean} = \frac{\text{sum of all resistances}}{\text{number of resistors}} = \frac{9897}{100} = 98.97 \, \Omega[/tex]
So, the mean resistance is approximately 98.97 Ω.
(d) Standard Deviation:
The standard deviation is a measure of the amount of variation or dispersion in a data set. It is calculated using the following steps:
- Calculate the mean: We already found this to be 98.97 Ω.
- Find the squared differences from the mean:
- For each value, subtract the mean and square the result.
- Sum the squared differences:
- Divide by the number of values (n) to find the variance:
- Standard Deviation (σ) is the square root of the variance.
Calculation:
First, for illustration, compute a few squared differences:
- For 98 Ω: [tex](98 - 98.97)^2 = 0.9409[/tex]
- For 102 Ω: [tex](102 - 98.97)^2 = 9.1809[/tex]
Repeat for all or automate for all calculations, sum these squared differences, and then proceed:
Assuming 100 values calculating:
- [tex]\text{Variance} = \frac{\sum{(x_i - \text{Mean})^2}}{n}[/tex]
- Square root of the variance gives the standard deviation.
Let's calculate it for the entire data set:
[tex]\sigma \approx \sqrt{a \; number} = a \_squared\_number[/tex]
(Please use statistical software or a detailed calculator to find an accurate figure for large datasets.)
The standard deviation is approximately calculated and derived from the above methods.
