data analysis 1

Measures of Central Tendency and Weighted Averages

The mean, median, and mode describe the center of a dataset. This lesson also explains how outliers can affect these summaries, introduces the range as a measure of spread, and shows why grades with different course-unit weights should be averaged using a weighted average.

Mean (Arithmetic Average)

The mean is calculated by adding all the values and dividing by the number of values.

For a sample, the mean is denoted by (\bar{x}):

[ \bar{x}=\frac{\sum_{i=1}^{n}x_i}{n} ]

For a population, the mean is denoted by (\mu):

[ \mu=\frac{\sum_{i=1}^{N}x_i}{N} ]

In both formulas, the summation means to add every observation. The values (n) and (N) represent the number of observations in the sample or population, respectively.

Example: For the scores (15, 16, 14, 19, 21, 23, 18, 14), the total is (140), so the mean is:

[ \frac{140}{8}=17.5 ]

The mean is sensitive to outliers—values unusually far from the rest of the data. Adding a score of (3) changes the mean to:

[ \frac{143}{9}\approx 15.9 ]

This illustrates how an extreme value can pull the mean downward.

Median

The median is the middle value in a dataset after the values have been ordered. Unlike the mean, it is less affected by extreme values.

To find the median:

1. Sort the observations from lowest to highest (or highest to lowest).
2. Find the middle position using ((n+1)/2).
3. If there is an odd number of observations, take the value in the middle position.
4. If there is an even number of observations, average the two middle values.

Examples:

• With the score of (3) added, the ordered data are (3, 14, 14, 15, 16, 18, 19, 21, 23). There are nine values, so the fifth value—and the median—is (16).
• In an even-sized example, if the two middle values are (15) and (16), the median is:

[ \frac{15+16}{2}=15.5 ]

Mode

The mode is the value that occurs most often in a dataset. A dataset may be:

• Unimodal: one mode
• Bimodal: two modes
• Trimodal: three modes
• Multimodal: more than three modes

In the lesson’s example, if (14) and (15) occur equally often and more than the other values, the data are bimodal. If (14) occurs more often than the other values, it is the single mode.

Range and Dispersion

Dispersion describes how spread out the data are. The range is a simple measure of dispersion:

[ \text{Range}=\text{highest value}-\text{lowest value} ]

Because the range depends only on the two extremes, it can be strongly affected by outliers.

Weighted Average

When observations have different levels of importance, a simple mean may not represent their contributions accurately. The lesson demonstrates this using course grades with different unit values.

The weighted-average formula is:

[ \text{Weighted average}=\frac{\sum x_iw_i}{\sum w_i} ]

Here, (x_i) is a grade and (w_i) is its weight, such as the number of course units.

Example: The grades and their units are (89) (2), (75) (3), (77) (4), (75) (5), (89) (3), and (95) (3).

The unweighted mean is:

[ \frac{500}{6}\approx 83.3 ]

For the weighted average, multiply each grade by its units, add the products, and divide by the total number of units:

[ \frac{89(2)+75(3)+77(4)+75(5)+89(3)+95(3)} {2+3+4+5+3+3} = \frac{1638}{20} = 81.9 ]

The results differ because courses with more units contribute more to the weighted average.

Grouped data will be discussed in another video.

Speakers and Sources

• One unnamed instructor or lecturer presents the lesson.
• No guests or external sources are identified in the subtitles.