Question

You have been observing the local Democratic Party in a large city and have compiled some information about a small sample of party regulars. Find the appropriate measure of central tendency for each variable. \(\begin{array}{cclcllc} \text { Respondent } & \text { Gender } & \text { Social Class } & \text { in Party } & \text { Education } & \text { Marital Status } & \text { No. of Children } \\ \hline \text { A } & \text { M } & \text { High } & 32 & \text { High school } & \text { Married } & 5 \\ \text { B } & \text { M } & \text { Middle } & 17 & \text { High school } & \text { Married } & 0 \\ \text { C } & \text { M } & \text { Working } & 32 & \text { High school } & \text { Single } & 0 \\ \text { D } & \text { M } & \text { Working } & 50 & \text { 8th grade } & \text { Widowed } & 7 \\ \text { E } & \text { M } & \text { Working } & 25 & \text { 4th grade } & \text { Married } & 4 \\ F & \text { M } & \text { Middle } & 25 & \text { High school } & \text { Divorced } & 3 \\ \text { G } & F & \text { High } & 12 & \text { College } & \text { Divorced } & 3 \\ \text { H } & F & \text { High } & 10 & \text { College } & \text { Separated } & 2 \\ \text { I } & F & \text { Middle } & 21 & \text { College } & \text { Married } & 1 \\ J & F & \text { Middle } & 33 & \text { College } & \text { Married } & 5 \\ \mathrm{~K} & \mathrm{M} & \text { Working } & 37 & \text { High school } & \text { Single } & 0 \\ \mathrm{~L} & \mathrm{~F} & \text { Working } & 15 & \text { High school } & \text { Divorced } & 0 \\ \mathrm{M} & \mathrm{F} & \text { Working } & 31 & \text { 8th grade } & \text { Widowed } & 1 \\ \hline \end{array}\)

Short answer

Gender: M (mode). Social Class: Working (mode). Time in Party: Mean ≈ 26.15, Median: 25, Mode: 25. Education: High school (mode). Marital Status: Married (mode). Number of Children: Mean ≈ 2.38, Median: 1, Mode: 0.

Step-by-step solution

Review the Variables

Identify the variables for which you need to find the measures of central tendency: Gender, Social Class, Time in Party (years), Education, Marital Status, Number of Children.

Determine the Measure for Gender

Gender is a categorical variable. The appropriate measure of central tendency is the mode. Count the frequency of each gender.

Calculate the Mode for Gender

There are 7 males (M) and 6 females (F). The mode is 'M' because it appears most frequently.

Determine the Measure for Social Class

Social Class is also a categorical variable. Calculate the mode by counting the frequency of each class (High, Middle, Working).

Calculate the Mode for Social Class

High: 3, Middle: 4, Working: 6. The mode is 'Working'.

Determine the Measure for Time in Party

Time in Party is a continuous variable. The appropriate measures are the mean, median, and mode.

Calculate the Mean for Time in Party

Sum the years: 32 + 17 + 32 + 50 + 25 + 25 + 12 + 10 + 21 + 33 + 37 + 15 + 31 = 340. Divide by the number of respondents: 340 / 13 ≈ 26.15 years.

Calculate the Median for Time in Party

Order the years: 10, 12, 15, 17, 21, 25, 25, 31, 32, 32, 33, 37, 50. The median is the middle value: 25.

Calculate the Mode for Time in Party

The mode is the most frequent value: 25 (appears twice).

Determine the Measure for Education

Education is a categorical variable. Calculate the mode.

Calculate the Mode for Education

High school: 6, College: 4, 8th grade: 2, 4th grade: 1. The mode is 'High school'.

Determine the Measure for Marital Status

Marital Status is a categorical variable. Calculate the mode.

Calculate the Mode for Marital Status

Married: 5, Divorced: 3, Single: 2, Widowed: 2, Separated: 1. The mode is 'Married'.

Determine the Measure for Number of Children

Number of Children is a discrete numerical variable. The appropriate measures are the mean, median, and mode.

Calculate the Mean for Number of Children

Sum the number of children: 5 + 0 + 0 + 7 + 4 + 3 + 3 + 2 + 1 + 5 + 0 + 0 + 1 = 31. Divide by the number of respondents: 31 / 13 ≈ 2.38 children.

Calculate the Median for Number of Children

Order the values: 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 4, 5, 7. The median is the middle value: 1.

Calculate the Mode for Number of Children

The mode is the most frequent value: 0 (appears 5 times).

Key concepts

categorical variables

Categorical variables are those that represent distinct groups or categories. These variables usually take on values that are names or labels. Examples include Gender, Social Class, and Marital Status. To summarize such data, we often use the mode, which is the category that appears most frequently. For instance, if we're looking at the 'Gender' variable in our dataset of Democratic Party members, we find that 'M' (Male) appears most often, making it the mode. Another example is 'Social Class', where 'Working' class appears the most frequently and is thus the mode.

continuous variables

Continuous variables can take on an infinite number of values within a given range. These variables are usually numerical and can be measured. In our sample, 'Time in Party' is an example of a continuous variable because it can take on any value within the range of the dataset. To measure central tendency for continuous variables, we often use the mean, median, and mode. These measures help summarize the data to give us an idea of the 'central' value. For example, to find the mean time in the party, we sum all the years and divide by the number of respondents, giving us a mean of approximately 26.15 years.

mode

The mode is the value that appears most frequently in a dataset. It's a measure of central tendency that is primarily used for categorical data but can also be used for numerical data. For example, if we look at 'Education' in our demographic sample, we find that 'High school' appears most often, making it the mode. The mode helps us understand the most common category or value in the dataset. In terms of 'Social Class,' the mode is 'Working' since it appears six times, more frequently than 'High' or 'Middle' classes.

mean

The mean, often called the average, is found by adding all the values in a dataset and then dividing by the number of values. It's a useful measure of central tendency for continuous variables. For example, to find the mean number of years that members have been in the party, we sum all the individual values: 32 + 17 + 32 + 50 + 25 + 25 + 12 + 10 + 21 + 33 + 37 + 15 + 31 = 340 years. We then divide by the number of members (13), yielding approximately 26.15 years. The mean helps us understand the central point around which the data is distributed.

median

The median is the middle value of a dataset when it is ordered from least to greatest. For continuous variables, the median is a very informative measure of central tendency, especially when the dataset has outliers. For instance, to find the median 'Time in Party,' we first arrange the years in ascending order: 10, 12, 15, 17, 21, 25, 25, 31, 32, 32, 33, 37, 50. Since we have 13 values, the seventh value, which is 25, is the median. The median provides a more accurate center of the dataset, especially if there are extreme values at either end.