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Basic Statistics Measures of Central Tendency

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STRUCTURE OF STATISTICS STATISTICS DESCRIPTIVE INFERENTIAL TABULAR GRAPHICAL NUMERICAL CONFIDENCE INTERVALS TESTS OF HYPOTHESIS

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Consider the following distribution of scores: How do the red and blue distributions differ? How do the red and green distributions differ?

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Characteristics of Distributions Location or Center –Can be indexed by using a measure of central tendency Variability or Spread –Can be indexed by using a measure of variability

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Consider the following distributions: How do they differ?

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Consider the following two distributions: How do the green and red distributions differ?

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Characteristics of Distributions Location or Central Tendency Variability Symmetry Kurtosis

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STRUCTURE OF STATISTICS STATISTICS DESCRIPTIVE INFERENTIAL TABULAR GRAPHICAL NUMERICAL CONFIDENCE INTERVALS TESTS OF HYPOTHESIS NUMERICAL

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STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES DESCRIPTIVE TABULAR GRAPHICAL NUMERICAL CENTRAL TENDENCY VARIABILITY SYMMETRY KURTOSIS

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Measures of Central Tendency Summarizing Data The Mean The Median The Mode Give you one score or measure that represents, or is typical of, an entire group of scores

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frequency score Most scores tend to center toward a point in the distribution. Central Tendency

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35 41 73 84 35 47 56 35 52 39 35 84 69 7735 47 92 35 33 43 47 65 39 90 49 35 67 The Mean The Median The Mode Frequency Tables Graphs Measures of Central Tendency 49 52 41 84 52 41 Measurement scales Frequency Tables & Graphs Averaging 52 43 47 Tabulating Graphing

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Measures of Central Tendency Are statistics that describe typical, average, or representative scores. The most common measures of central tendency (mean,median, and mode) are quite different in conception and calculation. These three statistics reflect different notions of the “center” of a distribution.

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“The Mode” The score that occurs most frequently In case of ungrouped frequency distribution

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When observations have been grouped into classes, the midpoint of the class with the largest frequency is used as an estimate of the mode. The mode of this distribution is estimated to be 52, the midpoint of the 51-53 class In case of grouped frequency distribution

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Unimodal Distribution -One Mode- Bimodal Distribution –Two Modes-

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Mode and Measurement Scales 1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2 3 2 1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4 112 132 112 113 112 150 125 114 68 56 39 56 44 56 45 56 75 81 67 59 Nationality 1=American 2=Asian 3=Mexican Football Poll 1=first 2=second 3=third 4=fourth IQ score Weight Can you find a mode for each data? 34 11256 Nominal ScaleOrdinal ScaleInterval Scale Ratio Scale

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It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. It can be found for ratio-level, interval- level, ordinal-level and nominal-level data “The Mode”

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The Median is the 50 th percentile of a distribution - The point where half of the observations fall below and half of the observations fall above In any distribution there will always be an equal number of cases above and below the Median. “The Median” Location Oh my !! Where is the median?

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For an odd number of untied scores (11, 13, 18, 19, 20) 11 12 13 14 15 16 17 18 19 20 The Median is the middle score when scores are arranged in rank order Median Location = (N+1)/2 = 3 rd Median Score = 18

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For an even number of untied scores (11, 15, 19, 20) 11 12 13 14 15 16 17 18 19 20 The Median is halfway between the two central values when scores are arranged in rank order Md score=(15+19)/2=17 Median Location = (N+1)/2 = 2.5 th

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The Median of group of scores is that point on the number line such that sum of the distances of all scores to that point is smaller than the sum of the distances to any other point. There is a unique median for each data set. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

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The Median can be computed for Ordinal-level data, or Interval-level data, or Ratio-level data.

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Median and Levels of Measurement 1 2 1 3 3 2 3 3 3 1 2 1 2 3 3 2 1 2 3 2 1 2 3 4 4 3 4 3 2 4 4 2 1 2 4 4 3 2 3 4 112 132 112 113 112 150 125 114 68 56 39 56 44 56 45 56 75 81 67 59 Nationality Football Poll IQ score Weight Can you find a median for each type of data? No Yes

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The Mean

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Definition: Definition: For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values. To compute the population mean, use the following formula. Population mean Sigma Population size Individual value

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Definition: Definition: For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values. To compute the sample mean, use the following formula. THE SAMPLE MEAN X-bar Sigma Individual value Sample Size

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Characteristics of The Mean Center of Gravity of a Distribution

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12345678 Mea n

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Data set 2527 31 29 3537 33 31 How much error do you expect for each case? The Mean 6 -6 -2 4 -4 0 2 Deviation Scores

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On average, I feel fine It’s too cold! It’s too hot!

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The Mean of group of scores is the point on the number line such that sum of the squared differences between the scores and the mean is smaller than the sum of the squared difference to any other point. If you summed the differences without squaring them, the result would be zero.

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Mean and Measurement Scales Every set of interval-level and ratio-level data has a mean. 1 2 3 Nationality 1=American 2=Asian 3=Mexican Nominal data 1 2 3 IQ Test Ordinal dataInterval dataRatio data Football Poll 1=first 2=second 3=third Weight 2 2 22 NOYES NO

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All the values are included in computing the mean.

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A set of data has a unique mean and the mean is affected by unusually large or small data values. 3579 The Mean 1 5 1 65 4 9 5 5.5 3 5

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Every set of interval-level and ratio-level data has a mean. All the values are included in computing the mean. A set of data has a unique mean. The mean is affected by unusually large or small data values. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

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The Relationships between Measures of Central Tendency and Shape of a Distribution

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Symmetric Unimodal Normal Distribution Mean=Median=Mode

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Positively Skewed Distribution Mode < Median < Mean Mode Median Mean The median falls closer to the mean than to the mode With unimodal curves of moderate asymmetry, the distance from the median to the mode is approximately twice that of the distance between the median and the mean

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Negatively Skewed Distribution Mode > Median > Mean Mode Median Mean The median falls closer to the mean than to the mode

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Bimodal Distribution Mean=Median Mode Mode 1 < Mean=Median < Mode 2

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If two averages of a moderately skewed frequency distribution are known, the third can be approximated. The formulae are: Mode = Mean - 3(Mean - Median) Mean = [3(Median) - Mode]/ 2 Median = [2(Mean) + Mode]/ 3

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Measures of Central Tendency as Inferential Statistics Parameters Statistics Sampling Mean Median Mode Difference Between Parameter and Statistics Sampling Errors

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As inferential measures, the Mean will be used much more frequently than the Median or Mode. Why ? On the average, there is less sampling error associated with the Mean than with the Median, and the Mode tends to have more sampling error than the Median. In other words, the difference between the statistic X and the Mean tends to be less than for the corresponding values for the sample Median (Md) and population median (Md pop ).

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SUMMARY There are three common measures of central tendency. The mean is the most widely used and the most precise for inferential purposes and is the foundation for statistical concepts that will be introduced in subsequent class. The mean is the ratio of the sum of the observations to the number of observations. The value of the men is influenced by the value of every score in a distribution. Consequently, in skewed distributions it is drawn toward the elongated tail more than is the median or mode. The median is the 50 th percentile of a distribution. It is the point in a distribution from which the sum of the absolute differences of all scores are at a minimum. In perfectly symmetrical distributions the median and mean have same value. When the mean and median differ greatly, the median is usually the most meaningful measure of central tendency for descriptive purposes. The mode, unlike the mean and median, has descriptive meaning even with nominal scales of measurement. The mode is the most frequently occurring observation. When the median or mean is applicable, the mode is the least useful measure of central tendency. In symmetrical unimodal distribution the mode, median, and mean have the same value.

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