2STRUCTURE OF STATISTICS TABULARDESCRIPTIVEGRAPHICALNUMERICALSTATISTICSCONFIDENCE INTERVALSINFERENTIALTESTS OF HYPOTHESIS
3Consider the following distribution of scores: How do the red and blue distributions differ?How do the red and green distributions differ?Red blue differ by location. We can index that by indicating the centers and how different they are.The red and green distributions differ by spread or variability of the data.
4Characteristics of Distributions Location or CenterCan be indexed by using a measure of central tendencyVariability or SpreadCan be indexed by using a measure of variability
5Consider the following distributions: How do they differ?They differ in symmetry.
6Consider the following two distributions: How do the green and red distributions differ?They differ in height or flatness or peakedness. This is called kurtosis.
7Characteristics of Distributions Location or Central TendencyVariabilitySymmetryKurtosis
8STRUCTURE OF STATISTICS TABULARDESCRIPTIVEGRAPHICALNUMERICALNUMERICALSTATISTICSCONFIDENCE INTERVALSINFERENTIALTESTS OF HYPOTHESIS
9STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES TABULARCENTRAL TENDENCYDESCRIPTIVEGRAPHICALNUMERICALVARIABILITYSYMMETRYKURTOSIS
10Measures of Central Tendency Summarizing DataThe MeanThe MedianThe ModeGive you one score or measure that represents, or is typical of, an entire group of scores
11Most scores tend to center toward a point in the distribution. frequencyscoreCentral Tendency
12Frequency Tables & Graphs Measures of Central Tendency 733352Averaging674335The MeanFrequency TablesTabulating84413947353552Graphing84493547The Median90355247Graphs435641843569Measurementscales3577The Mode3947659252414947
13Measures 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.
14“The Mode” The score that occurs most frequently In case of ungrouped frequency distribution
15When observations have been grouped into classes, the midpoint of the class with the largest frequency is used as an estimate of the mode.In case of grouped frequency distributionThe mode of this distribution is estimated to be 52, the midpoint of the class
16Unimodal Distribution -One Mode- Bimodal Distribution –Two Modes-
173 4 112 56 Mode and Measurement Scales Can you find a mode for each data?Nominal ScaleOrdinal ScaleInterval ScaleRatio Scale3411256Nationality1=American2=Asian3=MexicanFootball Poll1=first2=second3=third4=fourthIQ scoreWeight
18“The Mode”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
19“The Median” The Median is the 50th percentile of a distribution - The point where half of the observations fall below and half of the observations fall aboveIn any distribution there will always be an equal number of cases above and below the Median.Oh my !!Where is the median?Location
20Median Location = (N+1)/2 = 3rd For an odd number of untied scores (11, 13, 18, 19, 20)The Median is the middle score when scores are arranged in rank orderMedian Location = (N+1)/2 = 3rdMedian Score = 18
21For an even number of untied scores (11, 15, 19, 20) The Median is halfway between the two central values when scores are arranged in rank orderMedian Location = (N+1)/2 = 2.5thMd score=(15+19)/2=17
22The 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.
23The Median can be computed for Ordinal-level data, orInterval-level data, orRatio-level data.
24No Yes Yes Yes Median and Levels of Measurement NoYesYesYesNationalityFootball PollIQ scoreWeightCan you find a median for each type of data?
26Definition: 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.SigmaIndividual valuePopulation meanPopulation size
27THE SAMPLE MEANDefinition: 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.SigmaIndividual valueX-barSample Size
28Characteristics of The Mean Center of Gravity of a Distribution
29Center of Gravity of a Distribution 12345678Mean
30How much error do you expect for each case? Deviation Scores-62531-427313131312-231293133The Mean6435313137Data set
31On average,I feel fineIt’s too hot!It’s too cold!
32The 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.
332 2 2 2 YES NO YES NO Mean and Measurement Scales 1 2 3 1 2 3 1 2 3 Every set of interval-level and ratio-level data has a mean.Nominal dataOrdinal dataInterval dataRatio data2222YESNOYESNONationality1=American2=Asian3=MexicanFootball Poll1=first2=second3=thirdIQ TestWeight
34All the values are included in computing the mean.
35A set of data has a unique mean and the mean is affected by unusually large or small data values. 1133579955.556554The Mean
36Every 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.
37The Relationships between Measures of Central Tendency and Shape of a Distribution
39Positively Skewed Distribution ModeMedianMeanMode < Median < MeanThe median falls closer to the mean than to the modeWith 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
40Negatively Skewed Distribution ModeMedianMeanMode > Median > MeanThe median falls closer to the mean than to the mode
41Bimodal Distribution Mode1 < Mean=Median < Mode2 Mode Mode
42If 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]/ 2Median = [2(Mean) + Mode]/ 3
43Measures of Central Tendency as Inferential Statistics ParametersMean Median ModeDifference Between Parameter and StatisticsSamplingSampling ErrorsStatisticsMean Median Mode
44As 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 (Mdpop).
45SUMMARYThere 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 50th 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.