Question

For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand. \(\begin{array}{ll} {\text { Set 1 }} & {\text { Set 2 }} \\ \hline \text { a) } 4,7,7,7,10 & 4,6,7,8,10 \\ \text { b) } 100,140,150,160,200 & 10,50,60,70,110 \\ \text { c) } 10,16,18,20,22,28 & 48,56,58,60,62,70 \end{array}\)

Short answer

Set 2 for part a and c; Set 1 for part b have larger standard deviations.

Step-by-step solution

Analysis for Part a

For Set 1 (4, 7, 7, 7, 10), most numbers are concentrated around 7. In Set 2 (4, 6, 7, 8, 10), the numbers are more spread out across the range. Therefore, Set 2 likely has a larger standard deviation because the numbers are more dispersed.

Calculate Mean for Part a

For Set 1, the mean is \[ \frac{4 + 7 + 7 + 7 + 10}{5} = \frac{35}{5} = 7 \]For Set 2, the mean is \[ \frac{4 + 6 + 7 + 8 + 10}{5} = \frac{35}{5} = 7 \]Both sets have the same mean.

Calculate Variance for Part a

For Set 1, the variance is\[ \frac{(4-7)^2 + (7-7)^2 + (7-7)^2 + (7-7)^2 + (10-7)^2}{5} = \frac{9 + 0 + 0 + 0 + 9}{5} = 3.6 \]For Set 2, the variance is\[ \frac{(4-7)^2 + (6-7)^2 + (7-7)^2 + (8-7)^2 + (10-7)^2}{5} = \frac{9 + 1 + 0 + 1 + 9}{5} = 4 \]

Calculate Standard Deviation for Part a

For Set 1, the standard deviation is\[ \sqrt{3.6} \approx 1.90 \]For Set 2, the standard deviation is\[ \sqrt{4} = 2 \]Thus, Set 2 has a slightly larger standard deviation.

Analysis for Part b

Set 1 is (100, 140, 150, 160, 200), which is larger by a factor of 10 compared to Set 2 (10, 50, 60, 70, 110). The range and spacing are identical otherwise, so Set 1 will have a larger standard deviation.

Calculate Mean for Part b

For Set 1, the mean is \[ \frac{100 + 140 + 150 + 160 + 200}{5} = \frac{750}{5} = 150 \]For Set 2, the mean is \[ \frac{10 + 50 + 60 + 70 + 110}{5} = \frac{300}{5} = 60 \]Means are different, not affecting the standard deviation directly.

Calculate Variance for Part b

For Set 1, the variance is\[ \frac{(100-150)^2 + (140-150)^2 + (150-150)^2 + (160-150)^2 + (200-150)^2}{5} = \frac{2500 + 100 + 0 + 100 + 2500}{5} = 1040 \]For Set 2, the variance is\[ \frac{(10-60)^2 + (50-60)^2 + (60-60)^2 + (70-60)^2 + (110-60)^2}{5} = \frac{2500 + 100 + 0 + 100 + 2500}{5} = 104 \]

Calculate Standard Deviation for Part b

For Set 1, the standard deviation is\[ \sqrt{1040} \approx 32.25 \]For Set 2, the standard deviation is \[ \sqrt{104} \approx 10.20 \]Set 1 has a significantly larger standard deviation.

Analysis for Part c

Both sets have the same number of elements and range (Set 1: 10-28 and Set 2: 48-70), but Set 1 is more clustered around the mean compared to Set 2. Therefore, Set 2 likely has a larger standard deviation.

Calculate Mean for Part c

For Set 1, the mean is \[ \frac{10 + 16 + 18 + 20 + 22 + 28}{6} = \frac{114}{6} = 19 \]For Set 2, the mean is \[ \frac{48 + 56 + 58 + 60 + 62 + 70}{6} = \frac{354}{6} = 59 \]

Calculate Variance for Part c

For Set 1, the variance is\[ \frac{(10-19)^2 + (16-19)^2 + (18-19)^2 + (20-19)^2 + (22-19)^2 + (28-19)^2}{6} = \frac{81 + 9 + 1 + 1 + 9 + 81}{6} = 30.333 \]For Set 2, the variance is\[ \frac{(48-59)^2 + (56-59)^2 + (58-59)^2 + (60-59)^2 + (62-59)^2 + (70-59)^2}{6} = \frac{121 + 9 + 1 + 1 + 9 + 121}{6} = 43.667 \]

Calculate Standard Deviation for Part c

For Set 1, the standard deviation is \[ \sqrt{30.333} \approx 5.51 \]For Set 2, the standard deviation is\[ \sqrt{43.667} \approx 6.61 \]Hence, Set 2 has a larger standard deviation than Set 1.

Key concepts

Variance

Variance is a key concept in understanding how much your data spreads out or varies from the average value (the mean). It helps to quantify the extent of dispersion in a set of numbers. The larger the variance, the more spread out your data is.

To calculate variance, you first need to find the mean of your dataset. Then, for each number in the set, subtract the mean and square the result. Finally, average these squared differences to find the variance. The formula looks like this:

\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \]

Where:

• \( x_i \) is each individual data point,
• \( \bar{x} \) is the mean of the dataset,
• \( n \) is the number of data points.

Variance provides a foundational step for calculating the standard deviation, as it is simply the square root of the variance. This concept emphasizes the importance of both the mean and each data point's deviation from it. Understanding variance is crucial for a deeper grasp of statistical analysis.

Mean Calculation

Calculating the mean, or average, of a set of numbers is a simple yet fundamental step in many statistical methods. The mean provides a central value of the data from which deviations can be measured.

To find the mean, add up all the numbers in your dataset and then divide by the number of data points. The formula is straightforward:

\[ \text{Mean} = \frac{\sum x_i}{n} \]

Where:

• \( \sum x_i \) represents the sum of all data points,
• \( n \) is the total number of data points.

The mean helps to provide a quick snapshot of your data's center, making it a great starting point for further analysis like calculating variance or standard deviation. Keep in mind that the mean can be skewed by extreme values, which is why variance and standard deviation are also important to understand for a complete picture.

Statistical Analysis

Statistical analysis is an essential process that involves collecting and interpreting data to uncover patterns and trends. It helps in decision making across various disciplines like finance, psychology, and engineering.

An important aspect of statistical analysis is understanding measures of central tendency (like the mean) and measures of variability (like variance and standard deviation). These help to describe the data collectively and provide insight into the shape, spread, and average of the dataset.

Using statistical measures:

• The **mean** provides a central reference point.
• The **variance** shows how data points differ from the mean.
• The **standard deviation** offers a more intuitive measure of spread by taking the square root of variance, allowing for easier interpretation within the context of the data.

Statistical analysis goes beyond just crunching numbers; it involves critical thinking and interpretation of results to help explain real-world phenomena. This why each step, from the mean calculation to standard deviation, is crucial in understanding and leveraging the power of statistics.