## Which of the following are important properties of the arithmetic mean? Check all that apply.a. The mean can be calculated for nominal data.b. Σ(X-XX)=0 i.e. the sum of the deviations is zero.c. All of the values in the data are used in calculating the mean.d. The mean is always less than the median.e. There is only one mean for a set of data.

### Let’s go through each option and determine if it is an important property of the arithmetic mean:

a

Let’s go through each option and determine if it is an important property of the arithmetic mean:

a. The mean can be calculated for nominal data:

This statement is false. The arithmetic mean is used to calculate the average of numerical data. Nominal data consists of categories or labels and does not have a numerical value associated with it. Therefore, the mean cannot be calculated for nominal data.

b. Σ(X-XX)=0 i.e. the sum of the deviations is zero:

This statement is true. The arithmetic mean is determined by calculating the sum of all the values in the data set and dividing it by the total number of values. When calculating the mean, the deviations from the mean of each value cancel out and add up to zero.

c. All of the values in the data are used in calculating the mean:

This statement is true. The arithmetic mean takes into account all the values in the data set. Each value contributes to the calculation by being added together in the numerator.

d. The mean is always less than the median:

This statement is not always true. While it is true in some cases, it is not a universal property of the arithmetic mean. The relationship between the mean and median depends on the distribution of the data. In a skewed distribution, the mean can be influenced by extreme values and may be greater or less than the median.

e. There is only one mean for a set of data:

This statement is true. The arithmetic mean is a single value that represents the center of a data set. It is calculated using a specific formula and will result in only one mean for a given set of data.

So, the important properties of the arithmetic mean are:

– Σ(X-XX) = 0

– All of the values in the data are used in calculating the mean.

– There is only one mean for a set of data.

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