Statistics Week 2: Frequency Distributions
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Questions and Answers

What is the first task for a researcher after collecting data?

To organize and simplify the data

What is the goal of descriptive statistical techniques?

To simplify and organize data

What is the specific type of data organization called?

Frequency distribution

How can a frequency distribution be structured?

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

What are the two main elements of a frequency distribution?

<p>The original measurement scale and the number of individuals in each category (C)</p> Signup and view all the answers

What is the column in a frequency distribution table that lists the categories?

<p>X</p> Signup and view all the answers

What does the frequency (f) column in a frequency distribution table represent?

<p>How often each X value occurs in the data set</p> Signup and view all the answers

In a frequency distribution table, the values in the X column are listed from lowest to highest.

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

The sum of the frequencies in a frequency distribution table should equal N, which represents the total number of scores.

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

What does the proportion (p) in a frequency distribution table represent?

<p>The frequency divided by the total number of scores</p> Signup and view all the answers

Proportions are often referred to as relative frequencies because they describe the frequency in relation to the total number of scores.

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

How is the percentage in a frequency distribution table calculated?

<p>By multiplying the proportion by 100</p> Signup and view all the answers

The sum of the percentage column in a frequency distribution table should equal 100%.

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

What is a grouped frequency distribution?

<p>A frequency distribution table where the X column lists groups of scores called class intervals</p> Signup and view all the answers

What is the recommended number of class intervals for a grouped frequency distribution table?

<p>About 10 class intervals</p> Signup and view all the answers

What kind of number should the width of each class interval be?

<p>A relatively simple number</p> Signup and view all the answers

What should the bottom score of each class interval be?

<p>A multiple of the width</p> Signup and view all the answers

All intervals in a grouped frequency distribution table should have different widths.

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

What is the X axis of a frequency distribution graph used for?

<p>To represent the score categories</p> Signup and view all the answers

What type of graph is used for frequency distributions when the score categories consist of numerical scores from interval or ratio scales?

<p>All of the above (D)</p> Signup and view all the answers

What is the key characteristic of a histogram?

<p>A bar is centered above each score or class interval</p> Signup and view all the answers

What is the key characteristic of a polygon?

<p>A dot is centered above each score</p> Signup and view all the answers

How is a polygon completed?

<p>By drawing a line down the x-axis at each end of the range of scores</p> Signup and view all the answers

What kind of measurements are used when bar graphs are appropriate for frequency distributions?

<p>Measurements from a nominal or ordinal scale</p> Signup and view all the answers

What is the main difference between a histogram and a bar graph?

<p>There are gaps or spaces left between adjacent bars in a bar graph</p> Signup and view all the answers

When you can obtain an exact frequency for each score in a population, you can use the same frequency distribution graphs as for samples.

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

It is always possible to know the exact number of individuals, or the frequency, in a specific category for a population.

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

What can be used to showcase population distributions when the exact number of individuals is unknown?

<p>Relative frequency</p> Signup and view all the answers

What is the typical method for presenting population distributions measured on interval or ratio scales?

<p>A smooth curve</p> Signup and view all the answers

What is one commonly occurring population distribution?

<p>The normal curve</p> Signup and view all the answers

What do researchers commonly do to describe a frequency distribution?

<p>List its characteristics</p> Signup and view all the answers

What does central tendency measure?

<p>Where the center of the distribution is located</p> Signup and view all the answers

What does variability measure?

<p>The degree to which scores are spread over a wide range</p> Signup and view all the answers

What does a graph of a frequency distribution reveal about the data?

<p>The shape of the distribution</p> Signup and view all the answers

A symmetrical distribution implies that the left side of the graph is a rough mirror image of the right side.

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

What does it mean when a distribution is skewed?

<p>The scores tend to pile up toward one end of the scale</p> Signup and view all the answers

What is the section where the scores taper off toward one end of a distribution called?

<p>The tail</p> Signup and view all the answers

When a distribution is positively skewed, where do the scores pile up?

<p>On the left side of the distribution</p> Signup and view all the answers

What is an example of a positively skewed distribution?

<p>Wealth among citizens</p> Signup and view all the answers

What is a stem-and-leaf display?

<p>A method for obtaining and displaying a frequency distribution</p> Signup and view all the answers

What does the stem of a stem-and-leaf display consist of?

<p>The first digit or digits of the score</p> Signup and view all the answers

A stem-and-leaf display provides an organized picture of the entire distribution, allowing for identification of individual scores.

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

What is central tendency?

<p>A statistical measure that identifies a single score representing the center of a distribution</p> Signup and view all the answers

What is the goal of central tendency?

<p>To find the single score that is most typical or representative of an entire group</p> Signup and view all the answers

The average value obtained through central tendency can be used to describe an entire population or a sample.

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

Measures of central tendency can be used to compare groups of individuals or sets of data.

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

There's a single, standard procedure for determining central tendency.

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

What are the three methods commonly used for measuring central tendency?

<p>Mean, median, and mode</p> Signup and view all the answers

What is the mean?

<p>The sum of the scores divided by the number of scores</p> Signup and view all the answers

What is another way to think about the mean?

<p>As the amount each individual received when the total is divided equally.</p> Signup and view all the answers

What is the weighted mean?

<p>The overall mean for a combined group of scores</p> Signup and view all the answers

The weighted mean will always be halfway between the original two sample means if the two samples are of the same size.

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

The overall mean will always be halfway between the original two samples means unless the two samples have the same number of scores.

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

How is the weighted mean calculated?

<p>By summing together the sums of each group and dividing by the total number of scores in the combined group</p> Signup and view all the answers

How is the median calculated for an odd number of scores?

<p>The median is the middle score in the ordered list</p> Signup and view all the answers

Unlike the mean and median, the mode always corresponds to an actual score in the data.

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

What is a bimodal distribution?

<p>A distribution with two modes</p> Signup and view all the answers

What makes the mean a 'balance point'?

<p>The total distance above the mean is equal to the total distance below the mean</p> Signup and view all the answers

What is the best measure of central tendency to use when there are a few extreme scores?

<p>The median</p> Signup and view all the answers

The mean cannot be calculated if there is an undetermined value in the data.

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

A mean can be calculated for open-ended distributions where there is no upper or lower limit for one of the categories.

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

It is generally considered appropriate to use the mean as the measure of central tendency for ordinal data.

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

When is the mode a useful measure of central tendency?

<p>When the data is measured on a nominal scale</p> Signup and view all the answers

The mode can only be used as a measure of central tendency.

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

Measures of central tendency are commonly used in behavioural sciences to summarize and describe research results.

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

Measures of central tendency can be reported in text describing research results and presented in tables and graphs.

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

Line graphs, histograms, and bar graphs are all suitable for displaying means or medians in research reports.

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

The zero point of the x-axis and y-axis should always be at the point where the two axes intersect.

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

The height of a graph should ideally be at least two-thirds of the length of the graph for effective presentation.

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

The mean, median, and mode are all located at the center of a symmetrical distribution.

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

The mean, median, and mode are generally located at different points in a skewed distribution.

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

In a positively skewed distribution, the mode is usually the largest, followed by the median and then the mean.

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

In a negatively skewed distribution, the mean is usually the largest, followed by the median and then the mode.

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

Flashcards

Frequency Distribution

An organized tabulation that shows the frequency or count of individuals in each category on the scale of measurement. It can be presented as a table or a graph.

Frequency Distribution Table

A table containing at least two columns. The first column lists the categories on the scale of measurement (X), and the second column represents the frequency (f), showing how often each X value occurs in the dataset.

Frequency Distribution Total

The sum of the frequencies in a frequency distribution table should equal N, which is the total number of individuals in the dataset.

Proportion (p)

A third column in a frequency distribution table that shows the proportion (p) for each category, calculated by dividing the frequency (f) by the total number of individuals (N).

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Percentage (%)

A fourth column in a frequency distribution table showing the percentage of the distribution for each X value. It's calculated by multiplying the proportion (p) by 100.

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Grouped Frequency Distribution

A frequency distribution table where the X column lists groups of scores called class intervals instead of individual values. This is used when the data spans a wide range of values, preventing a list of individual values from being too long.

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Class Intervals

The groups of scores used in a grouped frequency distribution table. They are typically of a fixed width.

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Bottom Score

The lowest score in each class interval in a grouped frequency distribution table. These scores should be multiples of the interval width.

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Width (of a Class Interval)

The difference between the upper and lower limits of a class interval. All intervals should have the same width for consistency in a grouped frequency distribution.

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Histogram

A graphical representation of a frequency distribution where a bar is centered above each score or class interval. The height of the bar corresponds to the frequency, and the width extends to the real limits, so adjacent bars touch.

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Polygon

A graphical representation of a frequency distribution where a dot is centered above each score. The height of the dot corresponds to the frequency, and a continuous line connects the dots.

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Bar Graph

Used for nominal and ordinal scales. It's similar to a histogram but has gaps or spaces between adjacent bars. This separation emphasizes that the categories are distinct and not continuous.

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Relative Frequency Distribution

Shows the distribution of population data using relative frequencies instead of absolute counts when the exact number of individuals is not known.

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Smooth Curve

When scores in a population are measured on an interval or ratio scale, it's customary to represent the distribution as a smooth curve instead of a jagged histogram or polygon.

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Normal Curve

A commonly occurring population distribution defined by a specific shape that can measured by an equation.

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Central Tendency

A statistical measure that locates the center of a distribution. It helps identify a representative score for the entire group.

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Mean

The sum of all scores divided by the number of scores. It's represented by μ for a population and M for a sample.

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Median

The point in a distribution where half of the scores fall below and half fall above. It's useful for skewed distribution or when extreme scores are present.

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Mode

The score that occurs most frequently in a distribution. It's particularly useful for describing discrete variables and indicating the shape of the distribution.

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Bimodal

A distribution with two modes.

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Multimodal

A distribution with more than two modes.

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Mean and Extreme Scores

The mean can be affected by extreme scores or outliers, which makes it less representative of the overall distribution in such cases.

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Median and Skewed Distributions

When a distribution has a few extreme scores, the median is often used to report the average value because it's less sensitive to extreme scores.

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Median and Undetermined Values

The median is used when there are undetermined values in a dataset because it doesn't rely on the sum of the scores.

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Median for Ordinal Data

The median is typically the preferred measure of central tendency for ordinal data, as the mean may not be appropriate for such data.

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Mode for Nominal Data

Used for nominal scales, it identifies an actual score and offers insights into both central tendency and the shape of the distribution.

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Reporting Central Tendency

Measures of central tendency are used to summarize data and report results in research studies.

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Visualizing Central Tendency

Graphs are used to visualize and compare measures of central tendency, such as the mean or median.

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Central Tendency in Symmetrical Distributions

The mean, median, and mode are all located in the center of a symmetrical distribution.

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Central Tendency in Skewed Distributions

The mean, median, and mode are typically located in different positions in a skewed distribution, depending on the direction of the skew.

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Study Notes

Week 2: Frequency Distributions

  • Frequency distributions are used to organize and simplify data, allowing researchers to get a general overview of results.
  • Descriptive statistical techniques aim to achieve this goal.
  • A frequency distribution is a way to organize data, showing how many individuals fall into each category within a measurement scale.
  • It can be presented as a table or a graph.
  • The table/graph includes the set of categories from the original measurement and a record of the frequency (number of individuals) in each category.

Frequency Distribution Tables

  • Frequency distribution tables have at least two columns.
  • The first column lists categories/values within the measurement scale (X values) ranked from highest to lowest, without skipping values.
  • The second column lists the frequency (f) of each X value, represented by tallies that indicate how often each X value appears.
  • The sum of the frequencies should equal the total number (N) of data points.
  • An optional third column displays the proportion (p) for each category (calculated as f/N).
  • An optional fourth column displays the percentage corresponding to each X-value (calculated as p x 100).

Building a Frequency Distribution Table

  • Example: Using provided data points, the table shows how many times each score appears in the dataset (e.g., the number “8” appears 3 times).

Grouped Frequency Distribution

  • A grouped frequency distribution table is used when a set of scores spans a wide range.
  • It groups scores into class intervals (ranges of scores) instead of individual values, making the presentation easier to handle.
  • Ideally, it has approximately 10 class intervals.
  • Class interval widths should be a simple number (e.g., 2, 5, 10, or 20), and the bottom score of each interval should be a multiple of the width.

Example: Grouped Frequency Distribution

  • The provided example demonstrates a given dataset organized into a grouped frequency distribution table.

Frequency Distribution Graphs

  • Graphs are used to visually represent frequency distributions.
  • The x-axis represents the score categories (X values).
  • The y-axis represents the frequency.
  • When the score categories are numerical data (interval or ratio scale), a histogram or a polygon is used.

Histograms

  • Histograms use bars to represent scores, with each bar centered on a score (or class interval); bar height represents the frequency of that score (or class interval).
  • Bar widths extend to the real limits, with adjacent bars touching.

Polygons

  • Polygons use dots above each score to represent the frequency of the score.
  • Continuous lines connect the dots.
  • Lines extend down to the x-axis (zero frequency) at each end of the data range.

Bar Graphs

  • Bar graphs are used for nominal or ordinal scale data.
  • Bars represent categories, and gaps separate them; bar size represents the frequency.

Graphs for Population Distributions

  • In populations, where frequencies of every score are known, the graphs (histogram, polygon, bar graphs) replicate those for samples.
  • Populations are usually substantial that exact frequencies for categories often cannot be obtained.

Relative Frequencies

  • Relative frequencies, (proportion, p), are used to represent data when the exact population size isn't known.

Smooth Curve/Normal Curve

  • For interval/ratio data, distributions are often depicted by smooth curves rather than jagged histograms or polygons.
  • The normal curve is a common population distribution showing a specific, mathematically definable bell shape.

Example: Normal Distribution

  • Normal curves often exemplify IQ test results.

Describing Frequency Distributions

  • Important characteristics of a distribution are used to describe it, including central tendency, variability, and shape.

Central Tendency

  • Central tendency describes the central point of a distribution.
  • Mean, median, mode are the common measures used.

Shape

  • Shape describes how the scores are distributed in a graph.
  • Examples include symmetrical, skewed (positive and negative), unimodal, multimodal.

Positively and Negatively Skewed Distributions

Skewed distributions can be positively or negatively skewed.

Different Shapes for Distributions

  • Charts show examples of symmetrical and skewed distributions.

Stem-and-Leaf Displays

  • Stem-and-Leaf displays are a method for easily showing frequencies for a distribution.
  • The data is split into stems (left digits) and leaves (right digit).
  • Stems get a vertical line underneath to represent values.
  • Example uses of the method are given.

The Mean

  • The mean is the average of scores in a dataset.
  • It's calculated by adding all scores and then dividing by the total number of scores

Alternative Definitions of the Mean

  • Using the mean as the total amount each individual received when the total of scores (Σx) is divided equally among all individuals (N or n) in the distribution.
  • Mean as balance point of a distribution (total distance below the mean equals the total distance above the mean).

The Weighted Mean

  • The weighted mean is used to find an overall mean from several groups with different sizes.
  • It considers the size of each group when combining groups.

Characteristics of the Mean

  • Changes in scores typically change the mean.
  • Adding (or subtracting) the same constant to each score in a distribution will add a constant to (or subtract a constant from) the mean.
  • Multiplying (or dividing) each score in a distribution by a constant multiplies (or divides) the mean by a constant.

The Median

  • The median is the midpoint, the middle score, when data is ordered from smallest to largest.

Calculating the Median

  • Steps to determine the median, differing if the number of scores is odd or even.

The Mode

  • Mode is the score (if numerical) or category that occurs most frequently.

Bimodal/Multimodal

  • Some distributions are called bimodal or multimodal (more than one mode): where two or more categories have the greatest frequency

Selecting a Measure of Central Tendency

  • Which measure (mean, median or mode) is most appropriate depends on data distribution.
  • Skewed distributions are better represented by the median, and less affected (minimally impacted) by outliers.
  • Nominal data utilize the mode.

Extreme Scores or Skewed Distributions

  • When scores show substantial differences/extreme outlier values, the median is often a better central tendency measure.

Undetermined Values

  • If a score is unknown, the mean can't be calculated, but the median can still be determined.

Open-ended Distributions

  • In cases where some categories don't have upper or lower limits, the mean isn't determinable; median can be determined.

Ordinal Data

  • Many researchers believe that means should not be utilized when summarizing ordinal data. The median is usually the appropriate central tendency measure.

Reporting Measures of Central Tendency

  • Measures of central tendency are central to describing data from research reports.
  • Often, means, medians, and modes are shown in text and tables.
  • Graphs can also be used to describe and compare measures for central tendency.
  • Graphs should be effectively scaled (the height of the graph should approximate two-thirds to three-quarters its' length); the zero point of the axes is normally at the intersection point of the x and y axis, but can be adjusted as necessary to maximize viewing the data.

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Week 2 Frequency Distributions PDF

Description

This quiz on frequency distributions guides learners through organizing and simplifying data for statistical analysis. Participants will explore how frequency distribution tables are constructed, illustrating the relationship between categories and frequency counts. Gain insights into descriptive statistical techniques and their application in research.

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