You are working on a dataset. The recorded values are: 7, 9, 7, 7, 7, 6. Find the measures of central tendency and variability.

Steps to Solve

1. Calculate the mean

To find the mean (average) of the data set, sum all the values and divide by the number of values. The data set is: $7, 9, 7, 7, 7, 6$ $$ \text{Mean} = \frac{7 + 9 + 7 + 7 + 7 + 6}{6} = \frac{43}{6} \approx 7.17 $$

2. Calculate the variance

Variance measures how spread out the data is from the mean. First, find the squared difference of each value from the mean, sum these squared differences, and then divide by the number of values (or $n-1$ for sample variance). In the given prompt, it does not specify "sample," so we will assume it wants the population variance.

Calculate the squared differences: $(7 - 7.17)^2 \approx 0.0289$ $(9 - 7.17)^2 \approx 3.3489$ $(7 - 7.17)^2 \approx 0.0289$ $(7 - 7.17)^2 \approx 0.0289$ $(7 - 7.17)^2 \approx 0.0289$ $(6 - 7.17)^2 \approx 1.3689$

Sum of squared differences: $0.0289 + 3.3489 + 0.0289 + 0.0289 + 0.0289 + 1.3689 \approx 4.8529$

Divide by the number of values (6): $$ \text{Variance} = \frac{4.8529}{6} \approx 0.8088 $$

3. Calculate the standard deviation

The standard deviation is the square root of the variance: $$ \text{Standard Deviation} = \sqrt{0.8088} \approx 0.8993 $$

4. Ranking

To rank the data, sort the data and assign ranks: Sorted data: $6, 7, 7, 7, 7, 9$ Ranks: 6 (Rank 1) 7 (Rank 2.5, average of ranks 2, 3, 4, 5, i.e. $(2+3+4+5)/4 = 14/4 = 3.5$) 9 (Rank 6)

5. Questionnaires

Questionnaires are a method of gathering data from a sample of people. This is a qualitative or quantitative method of collecting data.