91Ó°ÊÓ

Here are the ages of 50 dancers who responded to a call to audition for a musical comedy: $$\begin{array}{llllllllll} \hline 21 & 19 & 22 & 19 & 18 & 20 & 23 & 19 & 19 & 20 \\ 19 & 20 & 21 & 22 & 21 & 20 & 22 & 20 & 21 & 20 \\ 21 & 19 & 21 & 21 & 19 & 19 & 20 & 19 & 19 & 19 \\ 20 & 20 & 19 & 21 & 21 & 22 & 19 & 19 & 21 & 19 \\ 18 & 21 & 19 & 18 & 22 & 21 & 24 & 20 & 24 & 17 \\ \hline \end{array}$$ a. Prepare an ungrouped frequency distribution of these ages. b. Prepare an ungrouped relative frequency distribution of the same data. c. Prepare a relative frequency histogram of these data. d. Prepare a cumulative relative frequency distribution of the same data. e. Prepare an ogive of these data.

Step by step solution

01

Compute Frequency Distribution

Firstly, count the number of times each age appears in the list. This gives the frequency distribution:

02

Compute Relative Frequency Distribution

For the ungrouped relative frequency distribution, divide the frequency of each age by the total number of ages. This will give a proportion or fraction for each age value. These values can also be converted to percentages for better interpretation.

03

Create a histogram

A histogram is a graphical display of the frequency distribution. The y-axis represents the frequency, while the x-axis represents the ages range. Each bar in the histogram represents the frequency of each age.

04

Compute Cumulative Relative Frequency Distribution

The cumulative relative frequency distribution represents the sum of the relative frequencies up to a certain point in the dataset. For each age, it is calculated as the sum of the relative frequencies of all ages up to and including that age.

05

Create an Ogive

An ogive is a type of line graph that shows cumulative frequencies. It has the same x-axis as the histogram, but the y-axis represents cumulative frequencies or cumulative relative frequencies. The points are plotted and then joined with a line. Each point on the graph represents the cumulative frequency up to that age.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relative Frequency Distribution

Understanding relative frequency can provide insight beyond just the raw counts of data. It measures how often a specific age appears compared to the entire dataset. To compute this, divide the frequency of each age by the total number of observations, which is 50 in this case. This yields a proportion or fraction for each age. For example, if the age 19 appears 15 times, its relative frequency is \( \frac{15}{50} = 0.3 \). This means that 30% of the dancers are 19 years old. Relative frequencies can easily be converted into percentages when needed.

• Relative frequencies help in understanding the spread of data.
• They show how each data point compares to the whole set.
• Useful in comparing data sets of different sizes.

Cumulative Frequency

Cumulative frequency provides a running total of frequencies, giving perspective on how data accumulates over a range. Each cumulative frequency is calculated by adding the frequency of the current category to the sum of frequencies of all preceding categories. For the age data, if frequencies for ages 17, 18, and 19 are 1, 3, and 15 respectively, the cumulative frequency for age 19 would be 1 + 3 + 15 = 19. These totals help show the number of observations below or equal to a certain age.

• Shows how data builds up over a range.
• Highlights the distribution of data points collectively.
• Helpful in identifying medians and percentiles.

Histogram

A histogram is a powerful graphical tool for displaying the distribution of data. It uses bars to represent the frequency or relative frequency of different age ranges in the dataset. The x-axis stands for the different ages or age ranges, while the y-axis tallies the frequency. Each bar's height reveals the age's frequency, offering a visual insight into where most data points cluster.

• Aids in visualizing frequency distributions clearly.
• Provides insights into data symmetry, skewness, and modality.
• Effective for making quick comparative analyses.

Ogive

An ogive is a cumulative frequency graph that aids in pinpointing trends and patterns over a continuous dataset. It is plotted with cumulative frequencies on the y-axis and the age classes on the x-axis. By connecting the points, you create a curve. This graph shows how data values add up across the distribution, often helping to identify medians and quartiles more visually.

• Useful for understanding cumulative data behavior.
• Helps in identifying key percentiles in the data quickly.
• Effective in analyzing cumulative growth or accumulation trends.