Final answer:
a. Variable: Days of drug use (interval level)
b. Mean: 7.64, Median: 6, Mode: 6, Variance: 25.73, Standard Deviation: 5.07, Range: 18
c. Standard Deviation (5.07): Indicates variability around the mean (7.64), Median (6): Half used drugs more, half less in the last month.
d. The median of 6 suggests that half of the sample used drugs for more than six days.
e. Prefer median for resilience to extreme values.
Step-by-step explanation:
a. Variable and Level of Measurement The variable in this scenario is the number of days a sample of drug addicts used drugs in the last 30 days. This variable is measured at the interval level, as it represents a quantitative measure with meaningful distances between values.
b. Calculations To calculate the mean, median, mode, standard deviation, variance, and range for the given data set: Days used drugs: 21, 11, 3, 9, 6, 9, 3, 6, 4, 6, 6
Mean (μ) = ΣX / N
= (21 + 11 + 3 + 9 + 6 + 9 + 3 + 6 + 4 + 6 + 6) / 11
= 84 / 11
= 7.64
Median = Middle value when the data is arranged in ascending order
Arranging the data in ascending order: 3, 3, 4, 6, 6, 6, 6, 9, 9, 11, 21
Median = (6 + 6) /2
=12/2
=6
Mode = The value that appears most frequently In this case, the mode is also 6 as it appears most frequently.
Variance (σ^2) = Σ(X - μ)^2 / N
= ((21-7.64)^2 + (11-7.64)^2 + (3-7.64)^2 + (9-7.64)^2 + (6-7.64)^2 + (9-7.64)^2 + (3-7.64)^2 + (6-7.64)^2 + (4-7.64)^2 + (6-7.64)^2 + (6-7.64)^2) /11
= (209.7156+10.9696+22.0996+1.3696+1.3696+1.3696+22.0996+1.3696+10.9696+1.3696+1.3696)/11
=283/11
≈25.73
Standard Deviation (σ) = √Variance
= √25.73
≈5.07
Range = Maximum value - Minimum value
=21 -3
=18
c. Interpretation of Standard Deviation The standard deviation measures the amount of variation or dispersion of a set of values from its mean.
In this case, the standard deviation of approximately 5.07 indicates that the number of days used drugs varies by around this amount from the mean of approximately 7.64.
d. Interpretation of Median The median represents the middle value when the data is arranged in ascending order and is not affected by extreme values or outliers in the data set.
In this case, the median of 6 suggests that half of the sample used drugs for more than six days and half for less than six days in the last month.
e. Shape of Distribution and Measure of Central Tendency The shape of the distribution can be determined by creating a histogram or using statistical tests for normality such as skewness and kurtosis calculations. If I could only select one measure of central tendency for this variable, I would choose the median because it is less affected by extreme values and provides a better representation of the typical number of days drug addicts used drugs in the last month.