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  5. Measures of Central Tendency

Measures Of Central Tendancy

Central tendency refers to the statistical measures that identify a single value representing the center or typical value of a dataset. It helps summarize large data sets with a single representative number.

Mean, Median, and Mode

  • Mean: The average of a data set, calculated by summing all values and dividing by the number of observations
  • Median: The middle value when data points are arranged in ascending order. If there's an even number of values, the median is the average of the two middle values
  • Mode: The most frequently occurring value in a dataset. A dataset can be unimodal (one mode), bimodal (two modes), or multimodal (multiple modes)

Positive and Negative Skews

Skewness describes the asymmetry in a dataset's distribution. It can be categorized in a couple of different ways:

  • Positive skew: The tail on the right side of the distribution is longer. The mean is greater than the median
  • Negative skew: The tail on the left side of the distribution is longer. The mean is less than the median

Outliers

Outliers are data points that significantly differ from the rest of the dataset. They can impact measures like the mean, making it less representative of the central value.

Weighted Mean

Weighted Mean accounts for different levels of importance among data points. It can be expressed algebraically as such:

\(\bar{x} \ = \cfrac{\sum x}{N}\)

Where:

  • \(\sum x \) is the sum of all values
  • \( N \) is number of values
  • \( \bar{x} \) is the weighted mean

The following are the ages of students in a class: 15, 17, 16, 19, 18, 17, and 16. Calculate the mean age of the students.

To find the mean, add all the ages together and divide by the total number of students:

\(\text{Mean} = \cfrac{15 + 17 + 16 + 19 + 18 + 17 + 16}{7}\)

\(= \cfrac{118}{7}\)

\(\approx 16.86\)

Therefore, we can determine that the mean age of the students is approximately 16.86.

The following scores were recorded in a math test: 45, 78, 92, 60, 85, 71, and 68. Find the median score.

To find the median, we can first arrange the scores in ascending order:

Sorted scores: 45, 60, 68, 71, 78, 85, 92

Next, we can find the middle value:

\(\text{Median} = 71 \; (\text{the middle value})\)

Therefore, we can determine the median score is 71.

In a survey of the number of books read by students, the following data was collected: 5, 7, 5, 8, 7, 6, 5, and 9. What is the mode of the number of books read?

The mode is the number that occurs most frequently. In this instance, we can determine that the mode is 5 since it occurs 3 times in the survey.

Calculate the mean number of hours studied by the group of students given the following table.
Hours Studied Number of Students
1 - 3 4
4 - 6 5
7 - 9 3
10 - 12 2

To find the mean, we can calculate the midpoint of each class interval, multiply by the number of students, sum the results, and divide by the total number of students.

We can let the midpoints be 2, 5, 8, and 11. We can then calculate the mean as such:

\(\text{Mean} = \cfrac{(2 \times 4) + (5 \times 5) + (8 \times 3) + (11 \times 2)}{4 + 5 + 3 + 2}\)

\(= \cfrac{8 + 25 + 24 + 22}{14}\)

\(= \cfrac{79}{14}\)

\(\approx 5.65\)

Therefore, we can determine the mean number of hours studied by the group of students as approximately 5.65.