Statistics: Normal Distribution Concepts
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What percentage of the area under the normal curve lies within one standard deviation of the mean?

  • 75.20%
  • 95.44%
  • 68.26% (correct)
  • 85.40%

    • For a normally distributed dataset, what percentage of cases lie within two standard deviations of the mean?

  • 95.44% (correct)
  • 99.72%
  • 68.26%
  • 90.00%

    • Which of the following Z scores corresponds to the percentage of area that covers 90% under the normal curve?

  • 2.00
  • 2.58
  • 1.96
  • 1.65 (correct)

    • In a sample of 1000 cases, approximately how many cases would fall within one standard deviation from the mean?

    <p>683</p> Signup and view all the answers

    What are Z scores primarily used for in a normal distribution?

    <p>To express original scores in units of standard deviation</p> Signup and view all the answers

    What is the mean of a standardized normal distribution?

    <p>0</p> Signup and view all the answers

    Which statement correctly describes Z scores?

    <p>Z scores can be interpreted as the number of standard deviations an individual score is from the mean.</p> Signup and view all the answers

    In the given example, what is the standard deviation of the raw scores 10, 20, 30, 40, 50?

    <p>14.14</p> Signup and view all the answers

    If a raw score is 30, what is its Z score in this example?

    <p>0.000</p> Signup and view all the answers

    What happens when raw scores are converted into Z scores?

    <p>They are transformed to a different scale with a mean of 0 and standard deviation of 1.</p> Signup and view all the answers

    What is the primary characteristic of the Normal Curve?

    <p>It is a symmetrical distribution.</p> Signup and view all the answers

    In what way does the Normal Curve serve empirical data?

    <p>It provides a theoretical model for describing distributions.</p> Signup and view all the answers

    Which of the following is NOT a property of the Normal Curve?

    <p>It can represent skewed data.</p> Signup and view all the answers

    What term is used to describe how well a statistical measure reflects the true value?

    <p>Validity.</p> Signup and view all the answers

    How is the Normal Curve visually represented?

    <p>As a perfectly smooth frequency polygon.</p> Signup and view all the answers

    In terms of measures of central tendency, which measure is NOT equivalent for a Normal Curve?

    <p>Range.</p> Signup and view all the answers

    What does it mean for a variable to be 'continuous'?

    <p>It can take on any value within a given range.</p> Signup and view all the answers

    Which method would best describe the characteristics of data in a Normal distribution?

    <p>Descriptive statistics that summarize data.</p> Signup and view all the answers

    What does a percentile indicate in a dataset?

    <p>The point below which a specific percentage of cases fall.</p> Signup and view all the answers

    In the process of finding a raw score from a percentile, what is the first step when given a percentile of 98.5%?

    <p>Subtract 50% from 98.5%.</p> Signup and view all the answers

    Which Z score corresponds to an area of 0.4850 as described in the example?

    <p>2.17</p> Signup and view all the answers

    How do you find the area between two scores that are on opposite sides of the mean?

    <p>Add the areas between each score and the mean.</p> Signup and view all the answers

    What is the combined area between IQ scores of 93 and 112?

    <p>36.25%</p> Signup and view all the answers

    When finding the area between two scores on the same side of the mean, what calculation is performed?

    <p>Subtract the smaller area from the larger area.</p> Signup and view all the answers

    If the area between Z scores of +0.65 and +1.05 is calculated to be 11.09%, what interpretation can be made?

    <p>11.09% of the total population falls between these scores.</p> Signup and view all the answers

    For a normal distribution, what does the mean represent?

    <p>The point where half of the data is below and half is above.</p> Signup and view all the answers

    What does the Z score represent in relation to a raw score?

    <p>The difference between the raw score and the mean, divided by the standard deviation.</p> Signup and view all the answers

    What is the probability of rolling a number other than 1 or 3 on your first try?

    <p>0.5000</p> Signup and view all the answers

    What type of variables are always discrete?

    <p>Nominal and ordinal variables</p> Signup and view all the answers

    In a discrete probability distribution, what should the sum of all probabilities equal?

    <p>1.00</p> Signup and view all the answers

    Which of the following best describes a continuous probability distribution?

    <p>Probabilities represent an area under the curve.</p> Signup and view all the answers

    How are probabilities calculated for continuous variables?

    <p>For a range of values under the curve</p> Signup and view all the answers

    Which of the following events has the same probability in rolling a die?

    <p>All outcomes have equal probabilities</p> Signup and view all the answers

    What is the probability of rolling a 2 or a 4 on a single roll of a die?

    <p>0.3334</p> Signup and view all the answers

    Which type of probability distribution requires calculations for a range of values due to their infinite nature?

    <p>Continuous distributions</p> Signup and view all the answers

    What is the probability of randomly selecting a child with an IQ between 95 and 100?

    <p>0.0987</p> Signup and view all the answers

    What is the probability of selecting a child with an IQ less than 123?

    <p>0.8749</p> Signup and view all the answers

    If 100 children are selected, approximately how many would have IQ scores less than 123?

    <p>88</p> Signup and view all the answers

    What can be inferred about cases that lie within three standard deviations of the mean?

    <p>The probability of selecting them is less than 0.0026.</p> Signup and view all the answers

    What summarizes the selection probability related to cases clustered around the mean?

    <p>Probability decreases with distance from the mean.</p> Signup and view all the answers

    What does the area under the normal curve indicate?

    <p>The probability distribution of continuous variables.</p> Signup and view all the answers

    What is the area representing the probability for cases within one standard deviation from the mean?

    <p>0.6826</p> Signup and view all the answers

    How does the probability of selecting a case compare when it is very close to the mean versus far from the mean?

    <p>Close scores have a higher selection probability.</p> Signup and view all the answers

    How many cases out of 10,000 would you expect to fall beyond three standard deviations from the mean?

    <p>2.6</p> Signup and view all the answers

    What is the significance of the Z score in determining probabilities?

    <p>It indicates the relationship between a score and the mean.</p> Signup and view all the answers

    What is the highest possible value for a probability according to the normal curve?

    <p>1.00</p> Signup and view all the answers

    If you want to find the probability of drawing a specific card from a deck, how would you express this probability?

    <p>0.0192</p> Signup and view all the answers

    Over an infinite number of draws, how many times would you expect to draw a king of hearts out of 10,000 draws?

    <p>192</p> Signup and view all the answers

    What is the total area under the normal curve that corresponds to a Z score of +0.85?

    <p>80.23%</p> Signup and view all the answers

    If a child scored 73, what percentage of children scored lower than this IQ score with a Z score of -1.35?

    <p>8.85%</p> Signup and view all the answers

    What does a probability value of 0.0192 indicate about the success of drawing a specific card?

    <p>It is unlikely to draw that card.</p> Signup and view all the answers

    What does a negative Z score indicate about a raw score in relation to the mean?

    <p>The raw score is below the mean.</p> Signup and view all the answers

    What is essential to determine the probability of an event in the context of a normal distribution?

    <p>The number of successes and events</p> Signup and view all the answers

    Why might the probability of drawing the king of hearts be considered improbable?

    <p>There are 52 other cards in the deck.</p> Signup and view all the answers

    What is the combined area below the mean represented by a Z score of -0.35?

    <p>63.68%</p> Signup and view all the answers

    When finding the area above a child's IQ score of 108, what is the corresponding proportion from the Z table?

    <p>65.54%</p> Signup and view all the answers

    What can be concluded about the nature of probabilities defined as successes over a large number of events?

    <p>They will maintain a certain proportional relationship.</p> Signup and view all the answers

    What percentage of area under the normal curve lies between +/- 2 standard deviations from the mean?

    <p>95.44%</p> Signup and view all the answers

    If the mean is 100, what is the raw score corresponding to a Z score of 1?

    <p>110</p> Signup and view all the answers

    What does a Z score of 0 indicate in a normal distribution?

    <p>The score is equal to the mean.</p> Signup and view all the answers

    Which Z score corresponds to the area that encompasses approximately 95% of the total area under the normal curve?

    <p>1.96</p> Signup and view all the answers

    In a dataset of 1000 cases, how many cases would fall within +/- 2 standard deviations from the mean?

    <p>950</p> Signup and view all the answers

    What is the first calculation step when determining the raw score from a given percentile?

    <p>Subtract 50% from the percentile</p> Signup and view all the answers

    When given two scores on opposite sides of the mean, how is the area between the scores calculated?

    <p>Add the areas between each score and the mean</p> Signup and view all the answers

    In the example provided, what is the calculated raw score for an IQ at the 98.5th percentile?

    <p>121.70</p> Signup and view all the answers

    What represents the proportion used in finding the Z score for the 98.5th percentile?

    <p>0.4850</p> Signup and view all the answers

    What signifies that 36.25% of the total area under the normal curve is between IQ scores of 93 and 112?

    <p>The sum of the individual areas from the mean to each score</p> Signup and view all the answers

    What would be the result when calculating the area on the same side of the mean for IQ scores of 113 and 121?

    <p>11.09%</p> Signup and view all the answers

    Which statement correctly describes the significance of percentiles in data analysis?

    <p>Percentiles indicate the positions of scores in relation to the total cases.</p> Signup and view all the answers

    What must be done when interpreting the area under the normal curve for cases between two scores?

    <p>Combine the individual areas from the mean</p> Signup and view all the answers

    Study Notes

    Quantitative Research Methods in Political Science

    • Lecture 4 focused on the Normal Curve and Z Scores.
    • Instructor: Michael E. Campbell
    • Course Number: PSCI 2702 (A)
    • Date: 09/26/2024

    Lecture Recap

    • Lecture 1: Covered the role of statistics in social sciences, use of systematic processes, difference between facts and values, characteristics of variables (discrete vs. continuous), and levels of measurement.
    • Lecture 2: Introduced causality (causal relationships), independent and dependent variables, conceptualization and operationalization, and instruments/instrumentation. Included discussion on systematic and random measurement error (reliability and validity).
    • Lecture 3: Recap of descriptive and univariate statistics. Discussed measures of central tendency (mode, median, mean). Covered measures of dispersion (e.g., IQR, variance, standard deviation). Also included frequency distribution tables and graphs/charts (pie, bar, histograms). All of this served as a foundation for the Normal Curve.

    The Normal Curve

    • A theoretical model in statistics.
    • Can be used to describe empirical distributions.
    • A perfectly smooth frequency polygon, unimodal (single mode/peak), and symmetrical (unskewed); mean, median, and mode are equivalent. (Healey, Donoghue, and Prus 2023, 126).
    • Shaped like a bell curve.
    • Tails extend infinitely.
    • Does not exist perfectly in nature. Empirical distributions will generally resemble this curve, but not match it perfectly.

    The Normal Curve Cont'd

    • Resembles the unskewed distribution from previous lecture.
    • Data in some empirical distributions are close to enough that we can treat them as Normal.

    The Normal Curve as a Tool

    • Used in descriptive statistics for statements about empirical distributions.
    • Useful for inferential statistics when generalizing from samples to populations.
    • Distances along the horizontal axis, when measured in standard deviations, always encompass the same proportion of the total area under the curve.
    • The distance between any point and mean cuts off the same proportion of the total area when measured in standard deviations.

    The Normal Curve Example

    • Data (IQ scores) for children and adults.
    • Distributions are symmetrical.
    • Each has a sample size of 1000.
    • Children: Mean = 100, Standard Deviation = 20
    • Adults: Mean = 100, Standard Deviation = 10

    The Normal Curve Example Cont'd

    • Larger data spread for children's IQ scores due to larger standard deviation.
    • Two scales: IQ units and standard deviations from mean.
    • No difference between these scales. Standard deviation units are a conversion.
    • One standard deviation above mean for children is 120, below mean is 80.

    The Normal Curve Example Cont'd: Adult IQ

    • Same logic applies to adult IQ scores
    • One standard deviation below the mean IQ for adults is 90, above is 110.

    Area Under the Normal Curve

    • When measured in standard deviations, the distances on the horizontal axis always encompass the same proportion of area under the curve.
    • +/-1 standard deviation = 68.26% of the area
    • +/-2 standard deviations = 95.44% of the area
    • +/-3 standard deviations = 99.72% of the area

    Z Scores (Standard Scores)

    • Z scores express scores after standardization to the theoretical normal curve.
    • Units are the standard deviation from mean
    • Original units can be anything (weight, time, IQ scores, etc.).
    • Z scores will always have a mean of 0 and standard deviations of 1

    Computing Z Scores

    • Equation to compute Z score: (individual score - sample mean) / sample standard deviation
    • Example using scores: 10, 20, 30, 40, 50. Need to calculate the mean then the standard deviaiton for these scores.

    Computing Z Scores Cont'd

    • Raw scores convert to Z scores, standardizing to mean of 0 and standard deviations of 1 for normal curve.
    • Example computation for the given scores.

    Positive and Negative Z Scores

    • Positive Z scores fall to the right of the mean.
    • Negative Z scores fall to the left of the mean. The absolute value of a Z-score indicates the distance from the mean in units of standard deviation.

    The Standard Normal Curve Table

    • A table that contains areas (probabilities) related to Z scores.
    • Found in most statistical textbooks.
    • Can be used to find the area between the mean and a given Z score; area beyond.
    • Area between Z and the mean or beyond a Z score.

    Finding Raw Scores

    • Percentile: a specific percentage of cases falling below a data point.
    • Using the normal curve, you can find raw scores given percentages of interest

    Finding Raw Score Example

    • Example using the 98.5th percentile of adult IQ scores.

    Finding the Area Between Two Scores on Opposite Sides of the Mean

    • Method for finding the area between two scores located on opposite sides of the mean.
    • Example using IQ scores 93 and 112.

    Finding the Area Between Scores on the Same Side of the Mean

    • Method for determining area between two similarly situated scores in relation to the mean.

    Using the Normal Curve to Estimate Probabilities

    • The Normal Curve can be used to estimate probabilities of events.
    • Probability: the likelihood of an event happening.

    Probabilities for Continuous Variables

    • Useful for figuring out the probabilities for continuous data, where we are looking at the ranges

    Probabilities for Continuous Variables Example

    • Example using the probability of randomly selecting a case from a normal distribution of children's IQ scores between 95 and 100.

    Probabilities for Continuous Variables Example #2

    • Example to determine the probability of randomly selecting a child with an IQ score less than 123.

    Probabilities at a Glance

    • High probability to select a case near the mean.
    • Low probability selecting a case far away from the mean.
    • The majority of cases cluster around the mean.

    Probabilities at a Glance Cont'd

    • Probability of selecting a case that falls beyond three standard deviations from the mean is very small.

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    Quantitative Research Methods In Political Science Lecture 4 PDF

    Description

    This quiz covers fundamental concepts related to the normal distribution, including Z scores, percentages of area under the curve, and characteristics of the normal curve. Test your understanding of how these statistical concepts apply in real-world datasets. Perfect for students of statistics looking to reinforce their knowledge.

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