Question

(a) Invent a sample of size 5 for which the deviations \(\left(y_{i}-\bar{y}\right)\) are -3,-1,0,2,2 (b) Compute the SD of your sample. (c) Should everyone get the same answer for part (b)? Why or why not?

Short answer

\(SD = 2.12\). As long as the same deviations and the mean are used, everyone will get the same answer.

Step-by-step solution

Create the Sample

To create a sample that fits the given deviations, you first need to decide on a mean value \(\bar{y}\) for the sample. After choosing a mean, add each of the deviations to this mean to get the actual sample values.

Choose a Mean and Calculate Sample

For simplicity, let's choose \(\bar{y} = 3\). Now, add the deviations to the chosen mean to get the sample: \((y_i - \bar{y}) = -3, -1, 0, 2, 2\) turns into \((y_1, y_2, y_3, y_4, y_5) = (0, 2, 3, 5, 5)\).

Calculate the Sum of Squares of Deviations

Sum the squares of the deviations \(\left(y_{i}-\bar{y}\right)^2\) and divide by the number of observations minus 1 for the sample variance: \[\frac{(-3)^2 + (-1)^2 + (0)^2 + (2)^2 + (2)^2}{5 - 1} = \frac{9 + 1 + 0 + 4 + 4}{4} = \frac{18}{4} = 4.5\]

Calculate the Standard Deviation

Take the square root of the sample variance to find the standard deviation (SD): \[SD = \sqrt{4.5} = 2.12\] (rounded to two decimal places).

Discuss Uniqueness of the Answer

Everybody must get the same answer for part (b) as the SD calculation is based strictly on the sample values and their deviations. Since these were specified, they lead to a unique value for standard deviation.

Key concepts

Sample Variance

Understanding the sample variance is a key element in statistics, as it measures how much the numbers in a set differ from the average (mean) of the sample. To find the sample variance, we perform the following steps:
First, calculate the mean of the sample by adding all numbers and dividing by the number of observations. Next, for each number, find the difference between it and the mean; this is known as the 'deviation from the mean'. Square each of these deviations to ensure that negative and positive deviations do not cancel each other out. The 'sum of squares' is the sum of these squared deviations. To find the variance, divide the sum of squares by the number of observations minus one (which is the sample size minus one).
In our example, with deviations of -3, -1, 0, 1, and 1 from the mean, and a sample size of 5, we calculate the variance as follows: \[\text{Sample Variance} = \frac{(-3)^2 + (-1)^2 + 0^2 + 2^2 + 2^2}{5-1} = \frac{9+1+0+4+4}{4} = 4.5\]This variance is a measure of the spread of the sample around its mean, which, in this case, is moderately dispersed.

Deviations from the Mean

When analyzing a data set, 'deviations from the mean' give us insight into the dispersion of the data points. These deviations represent the distance of each data point from the mean of the dataset. To determine these deviations, subtract the mean from each individual data point. Positive deviations indicate data points above the mean, while negative deviations indicate points below the mean.
If for example, our dataset includes 0, 2, 3, 5, 5 and the mean (\(\bar{y}\)) is 3, then the deviations are the differences: -3, -1, 0, 2, 2. These deviations are critical for calculating the sample variance and ultimately the standard deviation which quantifies the amount of variation in the dataset.

Sum of Squares

The 'sum of squares' is a concept frequently used in statistical calculations, including the computation of variance and standard deviation. It is the sum of the squared differences between each observation and the overall mean of the data. Squaring each deviation ensures that all values are positive and that we do not underestimate the variation due to canceling out of opposing numbers.
Using our sample dataset, where the deviations from the mean (\(\bar{y}\)) are -3, -1, 0, 2, 2, the sum of squares is calculated as \(((-3)^2 + (-1)^2 + 0^2 + 2^2 + 2^2) = (9 + 1 + 0 + 4 + 4) = 18\).This value is used as a stepping stone in the calculation of both the sample variance and the standard deviation, providing a numeric assessment of consistency within the set of data points.

Descriptive Statistics

Descriptive statistics are summary statistics that quantitatively describe or summarize features of a dataset. They include measures of central tendency such as mean, median, and mode, as well as measures of variability like range, variance, and standard deviation. Descriptive statistics provide a way to turn complex data into information that can be understood and used for analysis.
In the context of our exercise, to comprehend the dispersion and spread of our chosen sample, we use measures like the standard deviation, which was calculated from the sample variance. The standard deviation, in this case, is 2.12, after rounding to two decimal places, summarizes the average distance from the mean. Descriptive statistics play a crucial role in data interpretation, allowing us to move from raw data to actionable insights.