Final answer:
The relationship among mean, median, and mode in a symmetrical distribution is that they coincide at the same point. The mean tends to reflect skewing, and for skewed distributions, the mode is typically the largest in a left-skewed distribution and the mean the smallest. Which measure to use depends on the data's distribution shape.
Step-by-step explanation:
In statistics, when analyzing a symmetrical distribution, the mean, median, and mode all coincide at the same point. If the distribution is normal and symmetrical, these three measures of central tendency will be equal. When examining skewness in a distribution, the mean tends to reflect skewing the most because it is affected by outliers, whereas the median is a more robust measure that better represents the central tendency, especially in skewed data. To calculate the mean and standard deviation of the given data set (10; 11; 15; 15; 17; 22), one would add all numbers and divide by the total count for the mean, and use the sample standard deviation formula for the standard deviation.
For a data set exhibiting a left-skewed distribution, the mode will often be greater than the median, and the median will be greater than the mean, as the mean is pulled towards the longer, left tail. The question of which measure is more appropriate—mean, median, or mode—depends on the shape of the data distribution and whether it is skewed or symmetrical.