Final answer:
In a symmetrical distribution, the mean, median, and mode are equal. In skewed distributions, the mean is pulled towards the tail. The standard deviation determines the shape of the normal distribution curve, while the mean indicates its center line.
Step-by-step explanation:
In a symmetrical distribution the relationship among the mean, median, and mode is such that all three measures of central tendency are equal. This means that in a perfectly symmetrical distribution, often exemplified by the normal distribution, the mean, median, and mode are all located at the same central point. This central point usually corresponds to the highest peak of the distribution curve.
However, in skewed distributions, the mean is typically pulled towards the tail, and does not equal the median or mode. In a right-skewed distribution, the mean is greater than the median, which is greater than the mode. Conversely, in a left-skewed distribution, the mode is greater than the median, which is greater than the mean.
To calculate the mean and standard deviation from a given data set, one would sum up all the data values and divide by the number of values to find the mean. The standard deviation can then be calculated using the sample formula, which takes into consideration the variability of each data point from the mean.
When dealing with a normal distribution, the standard deviation plays a crucial role in determining the shape of the curve. If the standard deviation is changed, the curve becomes more spread out (larger standard deviation) or more peaked (smaller standard deviation). The mean marks the symmetry line of the distribution, and changing it shifts the entire curve to the left or right without altering its shape.