Question

statistics
mean, median, and mode: comparisons
2, 3, 6, 7, 8, 9, 11, 11, 12, 12, 13, 13, 13, 14, 14, 16, 17, 18, 19, 21, 23
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(a) which measures of central tendency do not exist for this data set? choose all that apply.
(b) suppose that the measurement 23 (the largest measurement in the data set) were replaced by 45. which measures of central tendency would be affected by the change? choose all that apply.

Explanation:

(a)

Step1: Recall definitions

Mean is \(\frac{\sum_{i = 1}^{n}x_{i}}{n}\), median is the middle - value (for \(n = 21\) data points, the \(11^{th}\) value when data is ordered), mode is the most - frequent value.

Step2: Check existence

For a numerical data set (all values are numbers), mean \(\bar{x}=\frac{2 + 3+6 + 7+8 + 9+11+11+12+12+13+13+13+14+14+16+17+18+19+21+23}{21}=\frac{276}{21}\approx13.14\), median is the \(11^{th}\) value (data is ordered: \(2,3,6,7,8,9,11,11,12,12,13,13,13,14,14,16,17,18,19,21,23\), so median \(=13\)), mode is \(13\) (appears \(3\) times).

(b)

Step1: Analyze effect on mean

Original mean \(\bar{x}_1=\frac{276}{21}\). New sum \(\sum x_2=276 - 23+45=298\), new mean \(\bar{x}_2=\frac{298}{21}\approx14.19\).

Step2: Analyze effect on median

Median is the \(11^{th}\) value. Since only the largest value is changed, the order of the first \(20\) values (for finding the \(11^{th}\) value) is not affected. So median remains \(13\).

Step3: Analyze effect on mode

Mode is still \(13\) (frequency of \(13\) is still \(3\), and no other value has a higher frequency).

Answer:

(a) All of these measures exist
(b) Mean