Descriptive Statistics: Central Tendency

Questions and Answers

What is the value of $n$ calculated from the frequency distribution?

5 (correct)

What is the sum of scores $Σx$ in the given data?

10 (correct)

What is the mean, denoted as $M$, in this frequency distribution?

8 (correct)

Under what condition is the median preferred as a measure of central tendency?

When there are extreme scores Signup and view all the answers

What is the mode of the distribution if it is unimodal based on the data provided?

3 Signup and view all the answers

Which statement is true regarding the median in skewed distributions?

It is a better measure than the mean Signup and view all the answers

Which of the following data types is most suitable for using the median?

Ordinal data Signup and view all the answers

What is the formula used to calculate the mean of a set of scores?

$X = \frac{\Sigma X}{n}$ Signup and view all the answers

Which measure of central tendency is defined as the score that occurs most frequently in a data set?

Mode Signup and view all the answers

In APA style, what notation is used to represent the size of the whole population?

N Signup and view all the answers

If the weighted mean is calculated using two samples with means 5 and 3, and sizes 4 and 6 respectively, what is the weight of the two samples combined?

10 Signup and view all the answers

What is the correct computation for the weighted mean when Sample 1 has a total of 20 and Sample 2 has a total of 18?

$\frac{38}{10}$ Signup and view all the answers

What is the weighted mean of the two hypothetical evaluations given?

3.85 Signup and view all the answers

What is the median of the following data set: 2, 2, 4, 5, 6?

4 Signup and view all the answers

In a bimodal distribution, which of the following is true?

There are exactly two modes. Signup and view all the answers

What is the mode of the data set: 2, 3, 5, 5, 6, 7, 9?

5 Signup and view all the answers

Which condition describes a negatively skewed distribution?

Mean < Median < Mode Signup and view all the answers

What is the halfway point in an even data set to find the median?

Average of the two middle numbers Signup and view all the answers

Which of these measures can be used for nominal scales?

Mode Signup and view all the answers

When calculating the weighted mean, why do we multiply the mean by the number of observations?

To find a representative total score. Signup and view all the answers

Study Notes

Central Tendency in Descriptive Statistics

Central tendency summarizes a set of scores with a single representative value.
Key measures include the mean, median, and mode.

Mean

Defined as the arithmetic average calculated by summing all scores and dividing by the number of scores.
Notation: Population mean (m, "mu") and sample mean (M or X).
Example calculation: Mean of the data set 2, 3, 3, 5, 6, 7, 9 results in 5.

Weighted Mean

Accounts for different group sizes when calculating an overall mean.
Formula: ( \text{Weighted Mean} = \frac{\Sigma X_1 + \Sigma X_2}{n_1 + n_2} ).
Instructors’ evaluations example resulted in a weighted mean of 3.85 from two groups.

Median

Represents the middle value of a data set when scores are arranged in ascending order.
For odd n, it's the central score; for even n, it's the average of the two middle scores.
Example: Median of 2, 2, 4, 5, 6 is 4; median of 2, 2, 3, 3, 5, 5 is also 4.

Mode

The most frequently occurring score in a data set.
Can apply to any measurement scale and is especially useful for nominal data.
Data set 2, 3, 5, 5, 6, 7 has a mode of 5; a bimodal distribution example is 2, 3, 5, 5, 5, 6, 7, 7, 7 with modes of 5 and 7.

Distributions

Unimodal distributions have one mode, while bimodal distributions have two.
Skewness affects how means relate to medians and modes:
- Symmetrical: Mean = Median = Mode
- Positively skewed: Mean > Median ≥ Mode
- Negatively skewed: Mean < Median ≤ Mode

Choosing a Measure

The mean is typically used for interval or ratio data, especially with symmetrical distributions.
The median is preferred for skewed data or when outliers are present.
Mode is suitable for nominal data and discrete variables.

Measures Summary

Mean: Likely preferred if distribution is normal.
Median: Preferred for skewed distributions or ordinal data.
Mode: Best for nominal data and determining the shape of distribution.

Application Example

To analyze a set of scores: construct a frequency table with relative frequencies, cumulative frequencies, and cumulative percents.
Example data analysis for scores results in:
- n = 21, Total Sum of Scores (SX) = 271, Mean = 12.90, Median = 13, Mode = 13, 14, 15.
- Indicates a slight negative skew with a distribution that is not dramatic.

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Description

Explore the concept of central tendency in descriptive statistics with this quiz. Learn about key measures such as mean, median, and mode that help summarize a data set. Test your understanding of how these measures represent a group's scores effectively.