[MUSIC] The mean, the median, and the mode are numerical measures which respond to the same questions as the graph, but numerically. They are concerning with the questions regarding the location of the center of the data set. That is, whether the data in the sample tend to be located At the center or located around a particular value. In other words, the measures of the central tendency provide information about what can be defined the typical observation in the data. At this stage in order to better understand the nature of this numerical measures, we need to keep in mind concept of parameter and the concept of statistic. As we've seen, a parameter refers to a specific population characteristic. A statistic refers to a specific sample characteristic. Now the measures of the central tendency are usually computed with respect to the sample rather than from the population data. The first measure of central tendency we look at is the one we use more often than we think in our everyday activities. This is the arithmetic mean, which normally is called mean or average. The arithmetic mean of a set of data is equal to the sum of the data values divided by the total number of the observation, N. If our data cover the whole population, then we have the population mean. The population mean is indicated with a Greek letter Me. Where N, the capital letter, is equal to the population size. If, instead, we use the sample data. Then we calculate the sample mean. The sample mean is indicated by the symbol X hat. The sample means statistic is given by the sum of the data values divided by the sample size n. The median is the middle observation of a set of observations that are arranged in increasing or decreasing order. In order to calculate the median, we need to look at the sample size and we need to find out if it is an odd number or an even number. In the first case, the median is the middle observation. Otherwise, the median is the average of the two middle observations. Summing up, for odd values, the median is the middle number. For even values, the median is the average of the two middle numbers. To find the median is it important to set the data either increasing or in a decreasing order. The median is located in the 0.50(n + 1) ordered position. If one mode exists in the data, then it will be the most frequently occurring value. A distribution can have one mode, then it's called unimodal. However, a distribution can show more than one mode, in that case, it will be called bimodal if it has two modes or multimodal for more than two modes. For example, let us find the mean of the following data 70, 94, 75, 77, 85, 82, 90, 95, 73, 92, 80, and 85. Describe the central tendency of the data. The average, or mean, is the sum of all the observations over the sample size, which is n = 12. And the mean is 83.17. In order to find the median we need to ordinate the data from the smaller to the largest value. And then we have 70, 73, 75, 77, 80, 82, 85, 85, 90, 92, 94, 95. The median is located into the 6.5 order position and (82+85)/2 = 83.5. Let us find the mode. As we have seen, the mode is the most frequently occurring value. And in our exercise the mode is equal to 85. Let us see another example. What is the mean, the median, and the mode for the following random sample percentage changes in interest paid in the current year With respect to the previous year, 0%, 0%, 8.1%, 13.6%, 19.4%, 20.7%, 10%, 14.2%. The mean percentage change in interest paid for sample is the sum of all the observations over the number of the observation, which is equal to 10.75%. The median percentage change in interest paid is equal to 11.8%. The mode is equal to 0% Since the value occurs twice. The other percentages instead occur only once. What we can notice here, is that the mode in the percentage rate does not represent the center of this sample. [MUSIC]