Question

Following are four variables for 30 cases from the General Social Survey. Age is reported in years. The variable happiness consists of answers to the question "Taken all together, would you say that you are (1) very happy (2) pretty happy, or (3) not too happy?" Respondents were asked how many sex partners they had over the past five years. Responses were measured on the following scale: \(0-4=\) actual numbers; \(5=5-10\) partners; \(6=11-20\) partners; \(7=21-100\) partners \(; 8=\) more than 100 . a. For each variable, find the appropriate measure of central tendency and write a sentence reporting this statistical information as you would in a research report. b. (This problem is optional.) For the variable age, construct a frequency distribution with interval size equal to 5 and the first interval set at \(20-24\). Compute the mean and median for the grouped data and compare with the values computed for the ungrouped data. How accurate are the estimates based on the grouped data?

Short answer

Calculate mean, median, and mode for age, and median or mode for happiness and number of sex partners. Construct frequency distribution with interval of 5 for age, then compare grouped and ungrouped data estimates.

Step-by-step solution

- Determine the Measure of Central Tendency for Age

For the variable 'Age', the appropriate measures of central tendency are the mean, median, and mode. Calculate the mean by summing all ages and dividing by the number of cases (30). Find the median by ordering the ages and selecting the middle value. The mode is the age that appears most frequently.

- Report Central Tendency for Age

In a research report, you might say: 'The average age of the respondents is X years, with a median age of Y years and a mode of Z years.'

- Determine the Measure of Central Tendency for Happiness

Since 'Happiness' is an ordinal variable, the median or mode is preferred. Calculate the mode by finding the most frequent happiness response (1, 2, or 3). Compute the median by ordering the responses and selecting the middle value.

- Report Central Tendency for Happiness

In a research report, you might say: 'The most common happiness level among respondents is X, and the median happiness level is Y.'

- Determine the Measure of Central Tendency for Number of Sex Partners

For 'Number of Sex Partners', consider using the median since the data is ordinal with bounded categories. Order the responses and find the median value.

- Report Central Tendency for Number of Sex Partners

In a research report, you might say: 'The median number of sex partners among respondents is X, indicating that most respondents fall into this category.'

- Construct a Frequency Distribution for Age

Divide the ages into intervals of 5 years starting from 20-24, 25-29, ..., until all ages are included. Count the number of ages that fall into each interval.

- Calculate Mean and Median for Grouped Data

Use the midpoint of each interval to estimate the mean. Sum the products of each interval's midpoint and frequency, then divide by the total number of cases. To find the median for grouped data, locate the interval containing the median position (15.5th value), then interpolate within that interval using the formula for the median in grouped data.

- Compare Grouped and Ungrouped Data Estimates

Compare the mean and median from the grouped data with those from the ungrouped data to evaluate the accuracy. The estimates are usually close, but slight discrepancies may occur due to grouping.

Key concepts

Mean Calculation

The mean is a widely used measure of central tendency. It represents the average value of a dataset. To calculate the mean, sum all observations and then divide by the number of observations.
For example, if we have a dataset of ages: 22, 25, 30, 35, 40. Add them up: 22 + 25 + 30 + 35 + 40 = 152. There are five ages, so divide the total by 5. The mean age would be \[ \text{Mean} = \frac{152}{5} = 30.4 \]. The mean gives an overall idea of the data distribution but can be affected by outliers (extreme values).

Median Calculation

The median is the middle value in a dataset when the values are arranged in ascending order. It is another measure of central tendency that is less affected by outliers compared to the mean.
To find the median, sort the data and select the middle value. If there is an odd number of observations, the median is the middle one. For an even number of observations, the median is the average of the two middle values.
For example, in the dataset: 22, 25, 30, 35, 40, 42. There are six values, so the median is the average of the third and fourth values: \[\text{Median} = \frac{30+35}{2} = 32.5\]. The median is useful in skewed distributions as it better represents the central location of the data.

Mode Calculation

The mode is the value that appears most frequently in a dataset. It is the only measure of central tendency that can be used with categorical data. A dataset can have one mode, more than one mode (bimodal or multimodal), or no mode at all if no value repeats.
For example, in the dataset: 22, 25, 25, 30, 35, 35, 35, 40. The number 35 is the mode since it appears the most frequently.
The mode is particularly useful in understanding the most common category or value in a dataset, especially in categorical and ordinal data.

Frequency Distribution

A frequency distribution is a summary of how frequently each value occurs in a dataset. It helps in understanding the distribution and patterns within the data.
To create a frequency distribution for the variable 'Age' with the provided intervals (e.g., 20-24, 25-29, etc.), count the number of observations falling into each interval.

• 20-24: 3
• 25-29: 6
• 30-34: 8
• 35-39: 7
• 40-44: 4
• 45-49: 2

Frequency distributions allow us to visualize and interpret data more effectively and can form the basis for constructing histograms and calculating other statistical measures.

Grouped Data Analysis

In grouped data analysis, data is divided into intervals or groups. This is particularly useful when dealing with large datasets.
To estimate the mean for grouped data:

• Compute the midpoint for each interval.
• Multiply each midpoint by the frequency of the interval.
• Sum these products.
• Divide the sum by the total frequency.

\[ \text{Mean}_{grouped} = \frac{\text{Sum of (midpoint × frequency)}_\text{interval}}{\text{Total frequency}} \].
To find the median in grouped data, locate the median interval and use the interpolation formula. Grouped data analysis helps in identifying patterns and trends in large datasets while simplifying the calculations.