Statistics Basic Definitions 2

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an organized tabulation of the number of individuals located in each category on the scale of measurement. ordered from highest to lowest, grouping together individuals with same score. provides an organized picture of the data. can be a table or a graph.

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Frequency Distribution

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It is possible to draw a vertical line through the middle so that one side of the distribution is a mirror image of the other.

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Symmetrical Distribution

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Scores tend to pile up toward on end of the scale and taper off gradually at the other end.

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Skewed Distribution

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Tail points toward the positive (above zero) end of x-axis

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Positively Skewed

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The mean is usually less than the median because the few low scores tend to shift the mean to the left.

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Negatively skewed

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Of a particular score is defined as the percentage of individuals in the distribution with scores equal to or less than a particular value.

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Percentile Rank

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When a score is identified by its percentile rank

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Percentile

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Statistical measure to determine a single score that defines the center of a distribution. The goal is to find the single score that is most typical or most representative of the entire group. Attempts to identify the average or typical individual.

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Central Tendency

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A distribution is the sum of the scores divided by the number of scores. Add all of the scores in the distribution and dividing by the number of scores. Can never be outside the range of scores.

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Mean

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Is the midpoint when scores in a distribution are listed from smallest to largest. Point on measurement scale below which 50% of the scores in the distribution are located.

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Median

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The score or category that has the greatest frequency in the distribution. Used to determine the typical or average value for any scale of measurement, including a nominal scale.

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Mode

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Extreme scores can influence the mean and not be representative of the distribution (median is not affected by extreme scores)

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Extreme Scores or Skewed distributions

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Impossible to compute the mean when there is an undetermined value (mean can still be determined though)

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Undetermined Values

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When there is no upper limit (or lower limit) for one of the categories. Can find a mean but can find the median.

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Open-ended Distributions

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Not appropriate to use the mean to describe central tendency for ordinal data. Median is always appropriate and is usually the preferred measure.

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Ordinal Scale

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Only option for describing cnetral tendency is to use the Mode

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Nominal Scales

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Because they only exist in whole, indivisble categories (when to use the Mode)

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Discrete Variables

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Because the mode requires little or no calculation, it is often included as a supplementary measure along with the mean or median. Identifies the location of the peak in the frequency distribution graph

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Describing Shape

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Provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together. Describes the distribution. Defined in terms of distance. Measures how well an individual score (or group of scores) represents the entire distribution.

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Variability

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Distance covered by the scores in a distribution, from the smallest score to the largest score. difference between the upper real limit and the lower real limit. Often, it does not give an accurate description of the variability for the entire distribution.

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Range

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Most commonly used and most important measure of variability. Provides a measure of the standard, or average, distance from the mean, and describes whether the scores are clustered closely around the mean or are widely scattered.

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Standard Deviation

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Equals the mean squared deviation. variance is the average squared distance from the mean.

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Population Variance

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Corrects for bias in sample variability.

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Degrees of Freedom

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If the average value of the statistic is equal to the population parameter. The average value of the statistic is obtained from all the possible samples for a specific sample size.

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Unbiased Sample Statistic

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If the average value of the statistic either understimates or overestimates the corresponding population parameter

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Biased Sample Statistic

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Adding a constant to each score does not change the standard deviation. Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant.

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Transformations of Scale

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Specify the precise location of each X value with in a distribution. The sign of the z-score (+/-) signifies whether the score is above the mean (pos) or below the mean (neg). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between x and mu.

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z-score

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Each z-score tells the exact location of the orginal X value within the distribution. The z-scores form a standardized distribution taht can be directly compared to other distributions that also have been transformed into z-scores

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Transforming z-scores

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The distribution will have exactly the same shape as the original distribution of scores.

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Shape of z-score distribution

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Distribution will always have a mean of zero.

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Mean of z-score distribution

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z-scores will always have a standard deviation of 1. Requires knowledge of the value of the population standard deviation (or variance)

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z-score distribution standard deviation

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used to test hypotheses about an unknown population mean when the value of sigma is unknown. Uses the estimated standard error in the denominator of the formula.

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t-statistic

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used as an estimate of the real standard error when the value of sigma is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample and the population mean

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estimated standard error

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The complete set of t values computed for every possible random sample for a specific sample size or a specific degrees of freedom. Approximates the shape of a normal distribution. The exact shape changes with degrees of freedom.

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t-distribution