Descriptive Statistics Overview
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Questions and Answers

Calculate the mean of the following data set: 2, 9, 11, 5, 6.

  • 6.6 (correct)
  • 5.8
  • 6.2
  • 7.2

    • Determine the median of the data set: 4, 8, 6, 1, 3.

  • 3
  • 6
  • 8
  • 4 (correct)

    • Calculate the mode of the following data set: 5, 2, 7, 2, 9, 2, 1.

  • 5
  • 7
  • 2 (correct)
  • 9

    • What is the position of the median in a dataset containing 15 values?

    <p>8th (B)</p> Signup and view all the answers

    In a data set with an even number of values, how is the median calculated?

    <p>The median is the average of the two middle values after sorting the dataset. (C)</p> Signup and view all the answers

    If the average height of women in a sample is 5'4" (64 inches) with a standard deviation of 3 inches, what is the z-score for a woman who is 5'7" (67 inches) tall?

    <p>1 (C)</p> Signup and view all the answers

    What is the probability of randomly selecting a faculty member who is 49 or older, given the following data: 9 faculty members are 41-45, 7 faculty members are 46-49, and 2 faculty members are 50 or older, with a total of 50 faculty members?

    <p>0.36 (A)</p> Signup and view all the answers

    In a set of measurements, what is the sum of all measurements divided by the total number of measurements called?

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

    According to Chebyshev's Theorem, what percentage of data will lie within 2 standard deviations of the mean?

    <p>75% (D)</p> Signup and view all the answers

    A numerical descriptive measure calculated for a sample is known as:

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

    A data set has a mean of 50 and a standard deviation of 10. What is the z-score for a value of 70?

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

    A data set has a mean of 100 and a standard deviation of 15. What is the range of values that would fall within 1 standard deviation of the mean?

    <p>85 to 115 (C)</p> Signup and view all the answers

    If the 25th percentile for a particular data set is 50, what percentage of the data lies above 50?

    <p>75% (B)</p> Signup and view all the answers

    The Empirical Rule states that approximately 68% of the data lies within one standard deviation of the mean. What percentage of the data lies beyond two standard deviations of the mean?

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

    A data set has a mean of 20 and a standard deviation of 5. If a value of 35 is observed, how many standard deviations away from the mean is this value?

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

    If a distribution is approximately mound-shaped, what percentage of the data falls within two standard deviations of the mean?

    <p>95% (A)</p> Signup and view all the answers

    What is the sample standard deviation for the following data: 5,12,6,8,14?

    <p>3.87 (B)</p> Signup and view all the answers

    Flashcards

    Tchebysheff’s Theorem

    A theorem stating that at least 1-1/k² of data values lie within k standard deviations of the mean.

    Empirical Rule

    A rule stating that for a normal distribution, about 68%, 95%, and 99.7% of data falls within 1, 2, and 3 standard deviations from the mean, respectively.

    Z-score

    A measure of how many standard deviations an element is from the mean of the data set.

    Standard Deviation

    A statistic that measures the dispersion of data from the mean.

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    Outlier

    A data point that is significantly different from other observations in a data set, typically having a z-score > 3 or < -3.

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    Percentile

    A measure indicating the value below which a given percentage of observations in a group of observations falls.

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    Mound-shaped distribution

    A type of data distribution that is symmetrical and bell-shaped, common for normal distributions.

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    Relative Standing

    The position of a particular measurement in relation to other measurements in the data set.

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    Population standard deviation (σ)

    A measure of the spread of a set of values in a population.

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    Sample standard deviation (s)

    A measure of the spread of a set of values in a sample.

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    Sample variance (s²)

    The average of the squared differences from the mean in a sample.

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    Why divide by n-1?

    Dividing by n-1 gives a better estimate of the population standard deviation.

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    k in Tchebysheff’s Theorem

    A number greater than or equal to 1 representing how many standard deviations from the mean.

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    Variability of data set

    A larger standard deviation or variance indicates greater spread in data values.

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    Skewed Right

    A distribution with a long tail on the right side, meaning more values are clustered on the left.

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    Parameter vs Statistic

    A parameter describes a population; a statistic describes a sample.

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    Arithmetic Mean

    The average calculated by summing all values and dividing by the count.

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    Finding the Mean

    Calculate mean using the formula: x = Σxi / n.

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    Median

    The middle value in a ranked dataset, calculated at position 0.5(n + 1).

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    Calculating Median Position

    Locate the median's position using 0.5(n + 1).

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    Mode

    The value that appears most frequently in a dataset.

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    Bimodal Distribution

    When two values appear most frequently in a dataset.

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    Study Notes

    Descriptive Statistics Overview

    • Descriptive statistics summarize and describe data.
    • Data can be either a sample or a population.
    • A variable is a characteristic that changes over time and/or between individuals.
    • Examples include hair color, temperature, account balance, number of students present in class.
    • An experimental unit is the individual or object on which a variable is measured.
    • A measurement is the actual value of the variable.
    • Data is a set of measurements.

    Types of Variables

    • Qualitative variables describe qualities, characteristics or categories.
    • Examples include hair color (e.g., brown, blonde, black), make of car (e.g., Dodge, Ford, Honda), gender (male, female), state of birth.
    • Quantitative variables describe numerical quantities.
    • Discrete quantitative variables can only assume certain values (and there are gaps between them).
    • Examples include the number of students in a class, number of cars in a parking lot.
    • Continuous quantitative variables can assume any value within a specified range.
    • Examples include height, weight, time.

    Data Distributions

    • A data distribution describes the values of a variable and how often each value occurs.
    • Frequency is how many times a value occurs.
    • Relative frequency represents the proportion or percentage of occurrences of a value.
    • Percent represents occurrences as a percentage of the total.

    Graphs for Univariate Data

    • Pie charts and bar charts are used to display qualitative data.
    • Line charts, bar charts, dotplots, stem-and-leaf plots, and relative frequency histograms are used to display quantitative data.

    Measures of Center

    • The mean, median, and mode describe the central tendency of a data set.
    • Mean: The average of a set of measurements. Calculation is the sum of measurements divided by the total number of measurements.
    • Median: The middle measurement when measurements are ranked from smallest to largest.
    • If the number of measurements is even, the median is the average of the two middle numbers.
    • Mode: The measurement that occurs most frequently in a data set.

    Measures of Variability

    • The range, variance, and standard deviation describe the spread or dispersion of a data set.
    • Range: The difference between the largest and smallest measurements.
    • Variance: The average of the squared deviations of the measurements about their mean.
    • Standard deviation: The positive square root of the variance.
    • Standard deviation is used to better understand the spread of the data distribution.

    Extreme Values and Outliers

    • Extreme values can significantly affect the mean, but not the median.
    • Outliers are markedly different from other measurements.

    z-Scores and percentiles

    • z-scores measure how many standard deviations a measurement is from the mean.
    • A z-score close to zero indicated a measurement around the mean.
    • A percentile indicates the percentage of measurements that are equal to or less than the measurements value.

    Box Plots

    • Box plots provide a visual representation of the distribution of data, including the median and quartiles.
    • The box in a box plot represents the interquartile range (IQR), which is the range encompassing the middle 50% of the data.
    • Outliers are clearly identifiable from the box plot.

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    Related Documents

    Chapter 2 Descriptive Statistics PDF

    Description

    This quiz provides a comprehensive overview of descriptive statistics, covering essential concepts such as variables, types of data, and measurement techniques. You'll learn the differences between qualitative and quantitative variables and gain foundational knowledge applicable in various fields. Test your understanding of how descriptive statistics summarizes data effectively.

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