Descriptive Statistics and Histograms

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Questions and Answers

What is the formula for calculating a z-score?

  • z = (x - xi) * s
  • z = xi * s
  • z = (xi - x) / s (correct)
  • z = (xi + x) / s

An outlier is defined as an observation that is always higher than the rest of the data.

False (B)

Calculate the z-score for a lizard running at a speed of 1.7 m/s given that x = 1.72 and s = 0.573.

−0.03

An observation that falls beyond Q3 + 1.5 × IQR or Q1 − 1.5 × IQR is known as an _____ .

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

Match the following statistics with their definitions:

<p>Mean = Average of all values Median = Middle value when ordered IQR = Difference between Q3 and Q1 Outlier = Value significantly different from others</p> Signup and view all the answers

How many data points fall within two standard deviations of the mean in Data Set 1?

<p>9 (A)</p> Signup and view all the answers

The Median and IQR are considered robust statistics.

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

What is the first step in calculating the p-th percentile from a sample?

<p>Order the data from smallest to largest.</p> Signup and view all the answers

What is the primary advantage of using standard deviation compared to variance?

<p>Standard deviation is in the same units as the original observations. (D)</p> Signup and view all the answers

The sample variance is denoted by 's'.

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

What does 's²' represent in statistics?

<p>Sample variance</p> Signup and view all the answers

The formula for sample variance is s² = (1/(n - 1)) ∑(x - x̄)², where x̄ is the ____.

<p>sample mean</p> Signup and view all the answers

Match the statistical term with its description:

<p>Interquartile Range = Difference between the first and third quartile Sample Mean = Average of a sample Variance = Average of squared differences from the mean Standard Deviation = Square root of variance</p> Signup and view all the answers

In the formula for sample standard deviation, how is 'n' determined?

<p>It represents the total number of observations. (C)</p> Signup and view all the answers

The interquartile range is calculated by subtracting the first quartile from the third quartile.

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

What is the formula to calculate the sample standard deviation using the variance?

<p>s = √(s²)</p> Signup and view all the answers

What is the first step in constructing a histogram for continuous data?

<p>Find the range of the data (C)</p> Signup and view all the answers

Histograms can be constructed using overlapping class intervals.

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

What percentage of earthquakes were recorded to be between 6.01 and 6.60?

<p>21%</p> Signup and view all the answers

Most intervals in a histogram should contain at least _____ measurements.

<p>5</p> Signup and view all the answers

Which of the following is NOT a requirement for class intervals in a histogram?

<p>They must be based on whole numbers only (C)</p> Signup and view all the answers

If the largest measurement is 8.1 and the smallest is 6.01, what is the range of the data?

<p>2.09</p> Signup and view all the answers

Match the class intervals with their frequency:

<p>6.01 – 6.30 = 12 6.31 – 6.60 = 18 6.61 – 6.90 = 15 6.91 – 7.20 = 9</p> Signup and view all the answers

To create a relative frequency histogram, it is necessary to round values to _____ decimal places.

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

What is the first step in calculating deviations from the mean?

<p>Add all scores together (B)</p> Signup and view all the answers

The mean of the test scores 72, 84, 96, 64, 88, 92, 74, and 78 is 81.8.

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

What do we do to eliminate the signs associated with deviations from the mean?

<p>Square the deviations</p> Signup and view all the answers

The sample standard deviation is calculated by taking the square root of the _____ of the squared deviations divided by n - 1.

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

Match the following terms with their definitions:

<p>Deviations from the mean = Differences found by subtracting the mean from each number Sample variance = Average of the squared deviations from the mean Sample standard deviation = Square root of the sample variance Mean = Average of all data points in a sample</p> Signup and view all the answers

Which formula correctly represents the calculation of the sample standard deviation?

<p>$s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n - 1}}$ (C)</p> Signup and view all the answers

The sample standard deviation can be a negative number.

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

How many observations are used when calculating the sample standard deviation of the given scores?

<p>8</p> Signup and view all the answers

What is the calculation used to find the mean of a data set?

<p>$ rac{x_1 + x_2 + ... + x_n}{n}$ (B)</p> Signup and view all the answers

A data set can have more than one mode.

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

What is the formula to find the range of a data set?

<p>Largest value - Smallest value</p> Signup and view all the answers

The value that occurs most often in a data set is called the _____ .

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

To find the median of an even-sized data set, you must:

<p>Average the two middle values (B)</p> Signup and view all the answers

List one measure of variation around the center.

<p>Variance, Standard Deviation, or Interquartile Range</p> Signup and view all the answers

What does the median represent in a data set?

<p>The middle value when the data is ordered (D)</p> Signup and view all the answers

Match the following statistical terms with their definitions:

<p>Mean = The average value of a data set Mode = The most frequently occurring value Median = The middle value of an ordered data set Range = Difference between the largest and smallest values</p> Signup and view all the answers

Flashcards

Histogram

A visual representation of data that uses bars to show the frequency of data values within specific intervals or categories.

Range of data

The difference between the largest and smallest values in a data set.

Class intervals

Dividing the range of data into equal intervals, each with a distinct value range.

Frequency

The number of data points that fall within a specific class interval.

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Relative frequency

The proportion of data points that fall within a specific class interval, expressed as a percentage.

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Relative frequency histogram

A histogram where the height of each bar represents the relative frequency of the corresponding class interval.

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Percentage of earthquakes between 6.01 and 6.6

The percentage of earthquakes with magnitudes between 6.01 and 6.6.

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Percentage of earthquakes greater than 6.9

The percentage of earthquakes with magnitudes greater than 6.9.

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Mean

The sum of all values divided by the number of values.

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Median

The middle value in a sorted dataset. If there are an even number of values, it's the average of the two middle values.

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Mode

The value that appears most frequently in a dataset. A dataset can have multiple modes or no mode at all.

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Range

The difference between the largest and smallest values in a dataset. It tells us the spread of the data.

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Measures of Center

Values that represent the center of a dataset, such as the mean, median, and mode.

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Measures of Variation

Values that describe the spread or variation of a dataset, such as range, variance, standard deviation, and interquartile range.

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Variance

A measure of how much individual data points vary from the mean. A higher variance indicates a greater spread of data.

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Standard Deviation

The square root of the variance. It represents the average distance of data points from the mean.

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Deviation from the mean

The difference between each data point and the average of the data set.

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Sum of Deviations from the Mean

The sum of all deviations from the mean will always be zero.

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Sample Variance

A measure of how spread out the data is from the mean, calculated by averaging the squared deviations.

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Sample Standard Deviation

The square root of the sample variance, providing a measure of the standard deviation from the mean.

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Calculating Sample Standard Deviation

The sample standard deviation is calculated by taking the square root of the sample variance.

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Standard Deviation as a Measure

It represents the typical difference between each data point and the average.

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Importance of Standard Deviation

A number that helps us understand the spread of the data.

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Squaring Deviations for Standard Deviation

It eliminates the negative signs by squaring the deviations, providing a more accurate measure of spread.

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What is the interquartile range (IQR)?

The interquartile range (IQR) represents the spread of the middle 50% of the data. It's calculated as the difference between the third quartile (Q3) and the first quartile (Q1).

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What is the first quartile (Q1)?

The 25th percentile, or Q1, is the value that separates the lowest 25% of the data from the rest.

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What is the second quartile (Q2)?

The 50th percentile, or Q2, is the median, which is the middle value of the data when it's ordered from smallest to largest.

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What is the third quartile (Q3)?

The 75th percentile, or Q3, is the value that separates the highest 25% of the data from the rest.

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What does the sample standard deviation (s) represent?

The sample standard deviation (s) measures how spread out the data is around the mean. A larger value of s indicates a greater spread, while a smaller value indicates data clustered closer to the mean.

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What is a z-score?

A z-score, or standardized score, measures how many standard deviations a data point is away from the mean. It tells us how far a data point is from the average relative to the data's variation.

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Z-score

A standardized score that indicates how many standard deviations an observation is from the mean.

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Outliers

Data points that fall significantly outside the typical range of values in a dataset, often more than 1.5 times the interquartile range (IQR) away from the quartiles.

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Interquartile Range (IQR)

The difference between the third quartile (Q3) and the first quartile (Q1) in a dataset.

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Robust Statistics

Statistical measures that are less affected by extreme values (outliers) in a dataset.

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Percentile

The value that represents the 100p-th percentile in a dataset, where 'p' is the proportion of data points less than or equal to that value.

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Sample Standard Deviation Formula

A statistical formula used to calculate the sample standard deviation. It involves summing the squared differences between each data point and the mean.

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First Quartile (Q1)

The value that separates the lowest 25% of the data from the rest.

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Third Quartile (Q3)

The value that separates the highest 25% of the data from the rest.

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Box Plot

A graphical representation of data that shows the median, quartiles, minimum, and maximum values. It helps visualize the distribution and spread of the data.

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Study Notes

Descriptive Statistics Handout 1

  • Histograms: Useful for displaying continuous data. They use relative frequencies (or percentages) to show distribution.
  • Data Construction: Histograms require intervals for values. Intervals should: not overlap, and have equal lengths, and contain at least 5 measurements.

Earthquake Magnitude Example

  • Data Range: Find the difference between the largest and smallest magnitudes.
  • Class Intervals: Divide the range into equal-size intervals (e.g., 6.01 - 6.30).
  • Frequency Table: Count the number of earthquakes in each interval.
  • Relative Frequency: Calculate the fraction (or percentage) of earthquakes within each interval relative to the total number of observations.
  • Examples include calculating the percentage of earthquakes between 6.01 and 6.60, percentage greater than 6.9, and those less than 7.21.

Categorical Data Example

  • Data Summary: Categorical data (like blood type) using frequency tables.
  • Relative Frequencies: Calculate the proportion (or percentage) of each category.
  • Histograms: A histogram displays the distribution from frequency or relative frequency tables

Measures of Center and Variation

  • Mean: The average of a set of data, calculated as the sum of the observations divided by the total number of observations. (xÌ„ = Σxáµ¢/n)
  • Median: The middle value in a sorted dataset. If there is an even number of data points, the median is the average of the two middle values.
  • Mode: The value that appears most often in a dataset. A dataset can have no mode or multiple modes.
  • Range: The difference between the largest and smallest values in a dataset.
  • Variance (s²): Measure of the spread of data points around the mean; calculated by summing the squared differences between each data point and the mean, then dividing by the number of observations minus one- (Σ(xáµ¢-xÌ„)²/(n-1))
  • Standard Deviation (s): The square root of the variance, providing a measure of the data dispersion on similar units.√(Σ(xáµ¢-xÌ„)²/(n-1))
  • A larger value for standard deviation indicates a greater dispersion of data.

Interquartile Range and Box Plots

  • Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1), representing the middle 50% of the data. IQR = Q3 - Q1
  • Box Plot: Illustrates the distribution of data using quartiles to show the median, IQR, and potential outliers.

Robust Statistics and the Median (Q2)

  • Robust Statistics: Less affected by outliers compared to mean and standard deviation.
  • Median: Middle value of a sorted set of numbers
  • IQR: Middle 50% of the data; resistant to outliers and good measure of spread when compared to the range.

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Module 1 Descriptive Statistics Handout 1 PDF

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