Considering that the confidence interval shown here includes the mean for the sample (61.8) and the mean for the population which we set apriori as 54, no significant difference is observed, between that which is expected and that which was observed. This estimate is illustrated in the following graph.

Calculate the Mean for A Frequency Distribution

In the following example, we compute the mean for frequency distribution. The formula to compute the mean of a frequency distribution is shown here as:

[latex]{\overline{x}} = {\Sigma{fx_i}\over{n}}[/latex]

Where:

• f refers to the frequency in each interval
• xi refers to the mid-point of the interval
• i refers to the “ith” case within the distribution
• n refers to all of the cases within the distribution.

Below is the frequency distribution table for the heights of 200 individuals. The data represent heights recorded in centimetres and organized into seven categories. The SAS code to compute the mean for this set of data is shown below the table. Notice that the table is reduced to a simple composition of two variables which includes the mid-point of the category represented by the variable: GRPMDPT, and the number of individuals, whose height scores fall within the specific category, represented by the variable: COUNTS.

The output generated by the SAS program above is the table of raw data presented in column form and includes the sums of the columns used to compute the mean for the frequency distribution.

* The mean is produced as the sum of the variable XP_RATIO

** The mean can also be calculated from the sum of the variable crsprdct ÷ 200

The Weighted Mean Score

In some situations, we may wish to combine means from several samples. Under such circumstances, we need to consider the sample size (or weight) of the distribution from which the means were drawn. By adjusting each independent sample mean by the number of subjects in the respective sample from which the means were drawn, we are able to provide different relative contributions of each mean to the total mean of all samples combined. The formula for a weighted mean from two samples is shown here. The formula for the mean of a sample is shown here:

[latex]{\overline{x}}={n_i\times{\overline{x_1}}+n_2{\overline{x_2}}\over{n_1 + n_2}}[/latex]

The Median Score

The median score is also a measure of central tendency, and it is defined as the middle score in a set of ordered scores.  In the example below, we begin with a set of scores (an array), we next sort the scores from lowest to highest.  Then we identify the number that is in the middle of the ordered set of scores where half the numbers are above the identified middle score, and half the numbers are below the identified middle score.

Example: Median

The median is the middle score. Considering the heart rate values again, we put these readings in order of magnitude and then identify which value is in the middle:

• 57
• 59
• 59
• 75
• 78
• 78
• 85
• 88
• 88
• 88

In this case, we have an even number of values (n = 10) so we can calculate the average of the two values in the middle. It just so happens that they are the same value in this example (78) so the median is 78.

• initial array of scores: {12, 72, 56, 34, 35, 13, 36, 16, 67}
• sorted array of scores: {12, 13, 16, 34, 35, 36, 56, 67, 72}
• sorted array of scores: {12, 13, 16, 34, 35, 36, 56, 67, 72}

Notice in the example above, regardless of the actual scores, the middle score in the ordered set of scores is the median, which in this set is 35.

When we have an even number of scores in our array there is a special caveat to identifying the median score in the distribution (set of scores). When we have two scores selected as the identified middle score we simply compute the average between the two identified middle scores and use that number as the median score.  That is, we add the two middle scores together and divide by 2.

• initial array of scores: {22, 32, 86, 44, 25, 13, 16, 18, 47, 11}
• sorted array of scores: {11, 13, 16, 18, 22, 25, 32, 44, 47, 86}
• computed median for the array: {11, 13, 16, 18, 22, 23.5, 25, 32, 44, 47, 86}

The Mode Score

The mode score is the third measure of central tendency, and it is defined as the most frequently occurring score in a set of scores. In the example below, we simply count the number of scores that are the same within a set of scores, within an array or within a distribution.

Below are 10 resting heart rate values:

78, 88, 57, 59, 75, 85, 88, 78, 59, 88

The mode is 88 because it appears most often.

In the following example of 16 scores, the number 2 occurs 3 times, but the number 27 occurs 4 times therefore we would identify 27 as the mode score.

2, 2, 2, 5, 6, 14, 15, 23, 26, 27, 27, 27, 27, 28, 37, 41