Descriptive Statistics. 2

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Questions and Answers

Why is it important to report the number of observations along with proportions and percentages?

  • It helps to impress readers with large numbers.
  • It allows readers to assess the adequacy of the sample size. (correct)
  • It guarantees that the results are universally applicable.
  • It reduces the need for statistical analysis.

What can be inferred if a report states, 'two out of three people questioned prefer courses in statistics' without mentioning the sample size?

  • The preference for statistics is uncommon.
  • The conclusion might be misleading due to a small sample. (correct)
  • The report is likely based on a large sample size.
  • The results are statistically significant.

Which formula should be used to calculate a proportion?

  • Divide f by n. (correct)
  • Subtract n from f.
  • Add f and n together.
  • Multiply f by n.

How is a percentage calculated from a proportion?

<p>By multiplying the proportion by 100. (A)</p> Signup and view all the answers

Which statement is true about the values of f and n in a dataset?

<p>f is usually smaller than n. (C)</p> Signup and view all the answers

What do ratios help to compare in a dataset?

<p>The relative frequency of categories of a variable. (B)</p> Signup and view all the answers

What is the maximum value that a proportion can have?

<p>1.00 (C)</p> Signup and view all the answers

What does a percentage of 43% imply regarding the sample size when referring to gender?

<p>43% of the sample consists of females out of the total sample size. (D)</p> Signup and view all the answers

What numerical code corresponds to individuals who identify as female?

<p>2 (C)</p> Signup and view all the answers

Which education level corresponds to the code 2?

<p>Community college (D)</p> Signup and view all the answers

Which age group had the youngest participant in the sample?

<p>23 (C)</p> Signup and view all the answers

What does a score of 1 in the 'Support for Gun Control' variable indicate?

<p>In favor of stricter gun laws (D)</p> Signup and view all the answers

How many different levels of education are represented by the numerical codes provided?

<p>4 (B)</p> Signup and view all the answers

What is the maximum age of participants in the provided cases?

<p>77 (B)</p> Signup and view all the answers

Which gender has the highest representation in the sample according to the cases provided?

<p>Male (C)</p> Signup and view all the answers

What numerical code represents individuals who have less than a high school education?

<p>0 (A)</p> Signup and view all the answers

What is the benefit of increasing the interval width in a frequency distribution?

<p>It allows for a more compact expression of data. (D)</p> Signup and view all the answers

Why are apparent gaps between intervals in a frequency distribution not problematic when age is measured in whole numbers?

<p>No data points fall between the intervals. (B)</p> Signup and view all the answers

What happens when age is measured with greater precision, such as in tenths of a year?

<p>It could create ambiguity in determining the interval for certain ages. (C)</p> Signup and view all the answers

What must be maintained to ensure mutual exclusivity between categories in a frequency distribution?

<p>Gaps and appropriate precision in defining intervals. (B)</p> Signup and view all the answers

If ages were recorded in whole numbers, what would happen if intervals were defined without gaps?

<p>Ambiguities in categorizing ages would arise. (A)</p> Signup and view all the answers

In the context of a frequency distribution, what defines an exhaustive and mutually exclusive set of intervals?

<p>Set limits at consistent levels of measurement and include gaps. (A)</p> Signup and view all the answers

How could the limits of age intervals be expressed if age is measured in tenths of a year?

<p>17.0–18.9 (B)</p> Signup and view all the answers

What is a consequence of not properly defining interval limits in a frequency distribution?

<p>Each case could fall into multiple intervals. (A)</p> Signup and view all the answers

What is the range of the given data set?

<p>34 (D)</p> Signup and view all the answers

Which interval represents the lowest scores in the frequency distribution?

<p>2–4 (A)</p> Signup and view all the answers

How many intervals should be used in the frequency distribution?

<p>10 (C)</p> Signup and view all the answers

What is the calculated interval size based on the range and number of intervals?

<p>3 (D)</p> Signup and view all the answers

What was the frequency of police response times of less than 6 minutes in 2000?

<p>275 (C)</p> Signup and view all the answers

Which time range saw the largest decrease in frequency from 2000 to 2020?

<p>Less than 6 minutes (D)</p> Signup and view all the answers

What is the highest interval in the frequency distribution represented?

<p>35–37 (C)</p> Signup and view all the answers

What percentage of the total response times in 2020 fall into the '21 minutes or more' category?

<p>5.0% (B)</p> Signup and view all the answers

How many cases fall within the interval of 14–16?

<p>17 (C)</p> Signup and view all the answers

What total number of students studied between 20 and 22 hours?

<p>15 (B)</p> Signup and view all the answers

How many people aged 40 or under reported having been in their mobile phone plan for 5 years?

<p>5 (B)</p> Signup and view all the answers

In the survey conducted by the telecommunications company, what is the cumulative percentage of responses for those over 40 with a plan duration of 2 years?

<p>17% (D)</p> Signup and view all the answers

What should a researcher do if the frequency distribution table seems unsatisfactory?

<p>Reconstruct it with different intervals (C)</p> Signup and view all the answers

What is the total frequency for police response times of 6–10 minutes in 2020?

<p>350 (D)</p> Signup and view all the answers

Which age group had the higher overall total in a surveyed mobile phone plan duration?

<p>Over 40 (B)</p> Signup and view all the answers

How many total responses were recorded in the year 2000 for police response times?

<p>940 (D)</p> Signup and view all the answers

What does the cumulative percentage column indicate about student study hours?

<p>55.24% of students studied less than 20 hours. (D)</p> Signup and view all the answers

How many intervals are suggested as a reasonable convention when constructing frequency distributions?

<p>10 to 15 intervals. (B)</p> Signup and view all the answers

Which step involves finding the range of scores?

<p>Subtracting the low score from the high score. (B)</p> Signup and view all the answers

What is the recommended way to define interval limits?

<p>Use the same level of precision as the data measurement. (A)</p> Signup and view all the answers

What should you do if the resultant frequency distribution has more than 15 intervals?

<p>It should be simplified for better comprehension. (A)</p> Signup and view all the answers

What does the calculation of interval size involve?

<p>Dividing the range by the number of intervals. (A)</p> Signup and view all the answers

How should frequencies be reported in a frequency distribution table?

<p>Frequencies should be reported in a separate column labelled 'Frequency'. (D)</p> Signup and view all the answers

What percentage of students studied between 17 and 19 hours?

<p>40.95% (D)</p> Signup and view all the answers

Flashcards

Sample size reporting

Always include the number of observations/subjects in a study alongside percentages when presenting data.

Percentages & proportions

Represent the relative size of a category within a dataset. Calculated by dividing the number of cases in a category by the total number of cases.

Levels of measurement

Percentages/proportions are valid for nominal & ordinal variables. They assess categorical data.

Calculating proportions

Divide the frequency of a category (f) by the total number of cases (n) in the dataset.

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Calculating percentages

Multiply the proportion calculated in a dataset by 100%.

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Ratio

A comparison of frequencies between categories of a variable, calculated by dividing the frequency of one category by another.

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Limit on proportions

Proportions cannot exceed 1.00 and percentages cannot exceed 100%.

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Data analysis

To accurately convey statistics, use percentages/proportions to evaluate relative proportions rather than using only raw counts.

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Interval Width

The range of values included in a data interval.

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Exhaustiveness

Every possible data point must fall into a category; no gaps in data coverage.

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Mutually Exclusive Categories

Data points can only belong to one category; no overlap between categories.

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Data Precision

The level of detail used to measure a variable (whole numbers, tenths, hundredths, etc.).

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Frequency Distribution

An organized table showing the number of data points in each category.

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Interval Limits

The minimum and maximum values defining the range of a data interval.

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Age Measurement Example

If measured in whole years, a 19-year-old belongs in the 19-20 age interval; more precise measurements demand interval limits with the same level of precision

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Ambiguity of intervals

When data is measured more precisely than the intervals, for example age is measured in fractions of years but the intervals are in single years, a case cannot be unambiguously assigned to an entry in the category.

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Interval Size

The width of each interval in a frequency distribution, calculated by dividing the range of scores by the desired number of intervals.

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Range of Scores

The difference between the highest and lowest score in a dataset.

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Cumulative Percentage

The percentage of cases that fall within a given interval and all intervals below it.

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Why use 10 intervals?

It's a common convention to use around 10 intervals in a frequency distribution for easier comprehension.

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Open-Ended Intervals

Intervals that have no specific upper or lower limit, used when data is extreme.

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Unequal Intervals

Intervals of different widths used when data is unevenly distributed.

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Non-Overlapping Intervals

Each case can only belong to one interval, preventing data from being counted twice.

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Clusterings in Data

Groups of data points that are close together in a frequency distribution, indicating potential patterns or trends.

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Arbitrary Decisions

Choices made in creating a frequency distribution that can affect the results, such as the number of intervals or the interval size.

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Frequency Distribution Table

A table that displays the frequency of observations within each interval, as well as percentages and cumulative percentages.

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Data Patterns

Recurring trends or relationships observed in a set of data, often revealed by a frequency distribution.

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Percentage

The relative size of a category expressed as a fraction of 100. Calculated by dividing the frequency of a category by the total number of cases and multiplying by 100.

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Variable

A characteristic or attribute that can vary among individuals or cases in a dataset.

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Categorical data

Data that can be grouped into distinct categories, such as gender, level of education, or support for stricter gun laws.

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Pie Chart

A visual representation of categorical data where proportions of each category are shown as slices of a circle.

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Bar Chart

A graph that uses bars to represent the frequency or percentage of each category of a variable.

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Construct a frequency distribution

To create a table that summarizes the data by showing the number of cases (frequency) for each category of a variable. This involves identifying the categories and calculating their frequencies.

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Response Time

The time taken for a police force to react to a call for assistance.

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Comparing response times

Analyzing changes in response times over different years to determine the efficiency of a police force.

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Study Notes

Descriptive Statistics: Percentages, Ratios, and Rates

  • Percentages and Proportions: Percentages standardize raw data to a base of 100, while proportions standardize to 1.00. These tools are used to contextualize research findings and make comparisons more meaningful.

  • Proportion Formula: Proportion (p) = f/n, where 'f' is the frequency in a category and 'n' is the total number of cases.

  • Percentage Formula: Percentage (%) = (f/n) x 100%, where 'f' is the frequency in a category and 'n' is the total number of cases.

Ratios

  • Ratio Definition: A ratio compares the relative frequency of two categories by dividing the frequency of one category by the frequency of another.

  • Ratio Formula: Ratio = f₁/f₂, where f₁ is the frequency of the first category and f₂ is the frequency of the second category.

Rates

  • Rate Definition: A rate expresses the relative frequency of a phenomenon over time or a defined period. Rates are calculated by dividing the number of actual occurrences (factual) by the number of possible occurrences (fpossible) in a specific time period.

  • Rate Formula: Rate = factual / fpossible

Frequency Distributions

  • Introduction: Frequency distributions summarize the distribution of variables by counting the number of cases in each category. They're useful for organizing and analyzing data.

  • Nominal Variables: Frequency distributions for nominal variables are straightforward: count occurrences in each category.

  • Ordinal Variables: Frequency distributions for ordinal variables are similar to nominal, but also include percentages which make the data more easily interpretable.

Charts and Graphs

  • Pie Charts: Pie charts visually represent percentages. A circle is divided into proportionally sized segments, each representing a category's percentage.

  • Bar Charts: Bar charts use bars of constant width to represent frequencies. Bar charts are particularly useful for showing comparisons between categories.

  • Histograms: Histograms represent continuous data. Areas of bars correspond to the frequency of occurrence within a specified interval.

Cumulative Frequency and Percentage

  • Cumulative Frequency: Shows the total number of cases below a particular score or range in an interval.

  • Cumulative Percentage: Represents the proportion of cases below a particular score or range.

Interval-Ratio Variables

  • Construction: Frequency distributions for interval-ratio variables typically involve larger numbers of scores requiring intervals/data grouping.

  • Midpoints: For certain tasks, the midpoint between the upper and lower limits of an interval is helpful.

  • Real Limits: Real limits account for the possibility a value could fall between interval boundaries.

###Unequal Intervals

  • Reasons for unequal intervals: Some values are more densely clustered within the intervals (or are outside of typical interval ranges). Using different intervals, or open-ended intervals, can make the data more easily interpreted.

SPSS Procedures

  • Frequency Distributions: Using SPSS, frequency distributions can be created and graphically visualized. The program provides cumulative frequency and percentage analysis.

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