Brief Summary
This video explains three measures of central tendency: mean, median, and mode. It details how to calculate each measure and how they are affected by outliers and transformations of the data, such as adding or multiplying by a constant. The video uses examples and Google Sheets to illustrate the concepts.
- The median is the middle value in an ordered data set, dividing it into a bottom 50% and a top 50%.
- The mode is the most frequently occurring value in a data set.
- Adding a constant to a data set shifts the median and mode by the same constant, while multiplying by a constant scales the median and mode by that constant.
Introduction to Median
The median is defined as the value that divides an ordered data set into the bottom and top 50%. To find the median, the data must first be arranged in increasing order. If the number of observations (n) is odd, the median is the value in the middle of the ordered list, specifically at position (n+1)/2. If n is even, the median is the average of the two middle values, located at positions n/2 and (n/2) + 1.
Calculating Median with Examples
To calculate the median, the data is first arranged in ascending order. For the data set 2, 12, 5, 7, 6, and 3, the ordered data is 2, 3, 5, 6, 7, 7, 12. With n = 7 (odd), the median is the (7+1)/2 = 4th observation, which is 6. A second data set, identical except for one value (105 instead of 12), yields the same median of 6, illustrating that the median is not sensitive to outliers. For a data set with an even number of observations, such as 2, 3, 5, 6, 7, and 105, the median is the average of the 3rd and 4th observations, (5+6)/2 = 5.5, which is not a member of the original data set.
Comparing Mean and Median
The mean is sensitive to outliers, while the median is not. The video uses Google Sheets to demonstrate calculating the median using the "median" function. For a given data set, the median is calculated to be 63.5. This is obtained by averaging the two middle values (61 and 66) since the number of observations is even (10).
Impact of Adding a Constant to the Median
Adding a constant to each value in a data set results in the new median being the old median plus that constant. For example, if a teacher adds 5 marks to every student's score, the new median is the old median plus 5. This is because adding a constant does not change the order of the observations.
Impact of Multiplying by a Constant to the Median
Multiplying each value in a data set by a constant results in the new median being the old median multiplied by that constant. For instance, if each mark is scaled down by a factor of 0.4, the new median is the old median times 0.4. The video demonstrates this with an example and verifies the results using Google Sheets.
Introduction to Mode
The mode is the value that occurs most frequently in a data set. If no value occurs more than once, the data set has no mode. In the data set 2, 12, 5, 7, 7, 6, 3, the mode is 7 because it appears twice.
Calculating Mode with Examples
Using the example data sets, the mode is calculated. For the data set 2, 12, 5, 7, 7, 6, 3, the mode is 7, as it appears twice. Similarly, for a second data set with a slight variation, the mode remains 7. However, a data set with all distinct values has no mode.
Impact of Adding or Multiplying by a Constant to the Mode
Adding a constant to each observation in a data set results in the new mode being the old mode plus that constant. Multiplying each observation by a constant results in the new mode being the old mode times that constant. The video uses examples and Google Sheets to illustrate these transformations.
Google Sheets Demonstration of Mode
The video demonstrates how to calculate the mode using the "mode" function in Google Sheets. It shows that adding a constant to the data set changes the mode by the same constant, and multiplying by a constant scales the mode by that constant.
Summary of Central Tendency Measures
The video summarizes the measures of central tendency: mean, median, and mode. It reviews the impact of adding a constant or multiplying by a constant on each of these measures.
