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Chapter 2: Problem 6

Why should relative frequencies be used when comparing two data sets?

Short Answer

Expert verified
Relative frequencies standardize the data, making comparisons between data sets of different sizes fair and valid.

Step by step solution

01

Understand relative frequency

Relative frequency measures how often something happens, divided by all outcomes. It is expressed as a percentage or a fraction of the total count. This allows comparisons between data sets with different sizes.
02

Identify data sizes

When comparing two data sets, check if their sizes (total number of observations) are the same or different. Recognize that different sizes can bias direct comparisons.
03

Calculate relative frequencies

For each data set, divide the frequency of each category by the total number of observations in that data set to get the relative frequencies.
04

Compare relative frequencies

Use the calculated relative frequencies to compare the categories across the two data sets. This way, you can directly compare how common an event is regardless of the data set sizes.
05

Interpret the results

Interpret the relative frequencies to understand the differences and similarities between the data sets. Highlight any significant patterns or outliers.

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Key Concepts

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

Data Comparison
When comparing two data sets, it's important not just to look at raw numbers. Raw counts can be misleading, especially if the data sets are of different sizes. Using relative frequencies helps provide a fair comparison.

Relative frequency measures how often a certain event occurs out of the total number of events. Comparing these frequencies gives a clearer picture of how events in different data sets relate to each other.

For example:
  • If one class has 30 students and another has 60, comparing the number of students who scored an A grade will be biased.
  • Instead, using the relative frequency (percentage of students with an A) makes the comparison fairer regardless of class size.
Data Set Size
Data set size plays a crucial role in any statistical analysis. Larger data sets can provide more accurate results simply due to the volume of data, but they can also introduce biases when compared directly to smaller data sets.

When you compare two data sets of different sizes, the larger one can dominate the results, making it appear that certain events are more or less common than they actually are.

For example:
  • If you have a survey of 1000 people showing 70% like ice cream, and another survey of 100 people showing 80%, the larger sample might seem more reliable. But without considering data set size, this direct comparison could be misleading.
Percentages and Fractions
To make comparisons easier, convert raw frequencies into relative frequencies using percentages or fractions. This process eliminates the influence of the total number of observations, allowing for a clearer comparison.

For example:
  • If Dataset A has 90 successes out of 300, the relative frequency is 30%.

  • If Dataset B has 80 successes out of 200, the relative frequency is 40%.

Despite Dataset A having more successes in absolute terms, Dataset B has a higher relative frequency, indicating a higher success rate.
Statistical Analysis
Statistical analysis with relative frequencies helps us see patterns, trends, and anomalies that might not be visible with raw data.

For example, in a survey of customer satisfaction across different regions, raw counts might show more satisfied customers in Region A simply because it has more respondents.

By converting to relative frequencies, we can identify which region has a higher percentage of satisfied customers, providing a clearer understanding of regional differences.

Relative frequencies also allow for more sophisticated analyses, such as identifying outliers or performing hypothesis testing. This makes it easier to make informed decisions based on the data.

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Most popular questions from this chapter

An exit poll was conducted in Los Alamos County, New Mexico, in which a random sample of 40 voters revealed whom they voted for in the presidential election. The results of the survey are as follows: $$\begin{array}{llll} \text { Kerry } & \text { Kerry } & \text { Bush } & \text { Bush } \\ \hline \text { Bush } & \text { Kerry } & \text { Kerry } & \text { Bush } \\ \hline \text { Kerry } & \text { Bush } & \text { Kerry } & \text { Bush } \\ \hline \text { Bush } & \text { Bush } & \text { Kerry } & \text { Kerry } \\ \hline \text { Kerry } & \text { Bush } & \text { Bush } & \text { Kerry } \\ \hline \text { Badnarik } & \text { Bush } & \text { Kerry } & \text { Bush } \\\ \hline \text { Kerry } & \text { Bush } & \text { Kerry } & \text { Bush } \\ \hline \text { Bush } & \text { Bush } & \text { Kerry } & \text { Kerry } \\ \hline \text { Bush } & \text { Bush } & \text { Bush } & \text { Nader } \\ \hline \text { Bush } & \text { Kerry } & \text { Bush } & \text { Kerry } \end{array}$$ (a) Construct a frequency distribution. (b) Construct a relative frequency distribution. (c) Construct a frequency bar graph. (d) Construct a relative frequency bar graph. (e) Construct a pie chart. (f) On the basis of the data, make a conjecture about which candidate will win Los Alamos County. Would your conjecture be descriptive statistics or inferential statistics? If George W. Bush wins Los Alamos County, what conclusions might be drawn, assuming that the sample was conducted appropriately? Would you be confident in making this prediction with a sample of \(40 ?\) If the sample consisted of 100 voters, would your confidence increase? Why?

A phlebotomist draws the blood of a random sample of 50 patients and determines their blood types as shown: $$\begin{array}{lllll} \hline O & O & A & A & O \\ \hline B & O & B & A & O \\ \hline A B & B & A & B & A B \\ \hline O & O & A & A & O \\ \hline A B & O & A & B & A \\ \hline O & A & A & O & A \\ \hline O & A & O & A B & A \\ \hline O & B & A & A & O \\ \hline O & O & O & A & O \\ \hline O & A & O & A & O \end{array}$$ (a) Construct a frequency distribution. (b) Construct a relative frequency distribution. (c) According to the data, which blood type is most common? (d) According to the data, which blood type is least common? (e) Use the results of the sample to conjecture the percentage of the population that has type O blood. Is this an example of descriptive or inferential statistics? (f) Contact a local hospital and ask them the percentage of the population that is blood type O. Why might the results differ? (g) Draw a frequency bar graph. (h) Draw a relative frequency bar graph. (i) Draw a pie chart.

Dividend Yield A dividend is a payment from a publicly traded company to its shareholders. The dividend yield of a stock is determined by dividing the annual dividend of a stock by its price. The following data represent the dividend yields (in percent) of a random sample of 28 publicly traded stocks of companies with a value of at least \(\$ 5\) billion. With the first class having a lower class limit of 0 and a class width of 0.40 (a) Construct a frequency distribution. (b) Construct a relative frequency distribution. (c) Construct a frequency histogram of the data. (d) Construct a relative frequency histogram of the data. (e) Describe the shape of the distribution. (f) Repeat parts (a)-(e) using a class width of 0.8. (g) Which frequency distribution seems to provide a better summary of the data? $$\begin{array}{lllllll} \hline 1.7 & 0 & 1.15 & 0.62 & 1.06 & 2.45 & 2.38 \\ \hline 2.83 & 2.16 & 1.05 & 1.22 & 1.68 & 0.89 & 0 \\ \hline 2.59 & 0 & 1.7 & 0.64 & 0.67 & 2.07 & 0.94 \\ \hline 2.04 & 0 & 0 & 1.35 & 0 & 0 & 0.41 \\ \hline \end{array}$$

Define raw data in your own words.

Violent Crimes Violent crimes include murder, forcible rape, robbery, and aggravated assault. The following data represent the violent crime rate (crimes per 100,000 population by state plus the District of Columbia in 2002 With the first class having a lower class limit of 0 and a class width of 150 (a) Construct a frequency distribution. (b) Construct a relative frequency distribution. (c) Construct a frequency histogram of the data. (d) Construct a relative frequency histogram of the data. (e) Describe the shape of the distribution. (f) Repeat parts (a)-(e) using a class width of 300. Which frequency distribution seems to provide a better summary of the data? (g) Do you believe that the violent crime rate is a good measure of how safe a state is? Why or why not? $$\begin{aligned} &\\\ &\begin{array}{llllll} 444 & 563 & 553 & 424 & 593 & 352 \\ \hline 311 & 599 & 1,633 & 770 & 459 & 262 \\ \hline 255 & 621 & 357 & 286 & 377 & 279 \\ \hline 662 & 108 & 770 & 484 & 540 & 268 \\ \hline 343 & 539 & 352 & 314 & 638 & 161 \\ \hline 375 & 740 & 496 & 470 & 78 & 351 \\ \hline 503 & 292 & 402 & 285 & 822 & 177 \\ \hline 717 & 579 & 237 & 107 & 291 & 345 \\ \hline 234 & 225 & 274 & & & \\ \hline \end{array} \end{aligned}$$
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