Descriptive Descriptive StatisticsStatistics

Research WritingResearch WritingAiden Yeh, PhDAiden Yeh, PhD

Descriptive statistics is the term given to the Descriptive statistics is the term given to the analysis of data that helps describe, show or analysis of data that helps describe, show or summarize data in a meaningful way such tsummarize data in a meaningful way such that, for example, patterns might emerge frohat, for example, patterns might emerge from the data. m the data.

Descriptive statistics do not, however, allow Descriptive statistics do not, however, allow us to make conclusions beyond the data we us to make conclusions beyond the data we have analysed or reach conclusions regardinhave analysed or reach conclusions regarding any hypotheses we might have made. They g any hypotheses we might have made. They are simply a way to describe our data. are simply a way to describe our data.

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For example, if we had the results of 1For example, if we had the results of 100 pieces of students' coursework, we 00 pieces of students' coursework, we may be interested in the overall performay be interested in the overall performance of those students. We would alsmance of those students. We would also be interested in the distribution or spo be interested in the distribution or spread of the marks. Descriptive statisticread of the marks. Descriptive statistics allow us to do this. s allow us to do this.

Using statistics and graphsUsing statistics and graphs

Frequency DistributionFrequency Distribution The distribution is a summary of the frequency oThe distribution is a summary of the frequency o

f individual values or ranges of values for a variaf individual values or ranges of values for a variable. The simplest distribution would list every vable. The simplest distribution would list every value of a variable and the number of persons who lue of a variable and the number of persons who had each value. For instance, a typical way to deshad each value. For instance, a typical way to describe the distribution of college students is by yecribe the distribution of college students is by year in college, listing the number or percent of stuar in college, listing the number or percent of students at each of the four years. Or, we describe gdents at each of the four years. Or, we describe gender by listing the number or percent of males ender by listing the number or percent of males and females. In these cases, the variable has few and females. In these cases, the variable has few enough values that we can list each one and sumenough values that we can list each one and summarize how many sample cases had the value. marize how many sample cases had the value.

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Distributions may also be displayed usiDistributions may also be displayed using percentages. For example, you coulng percentages. For example, you could use percentages to describe the:d use percentages to describe the: percentage of people in different income lpercentage of people in different income l

evelsevels percentage of people in different age rangpercentage of people in different age rang

eses percentage of people in different ranges opercentage of people in different ranges o

f standardized test scoresf standardized test scores

Measures of central tendency:Measures of central tendency: these are these are ways of describing the central position of ways of describing the central position of a frequency distribution for a group of daa frequency distribution for a group of data. In this case, the frequency distributiota. In this case, the frequency distribution is simply the distribution and pattern of n is simply the distribution and pattern of marks scored by the 100 students from thmarks scored by the 100 students from the lowest to the highest. We can describe te lowest to the highest. We can describe this central position using a number of stahis central position using a number of statistics, including the mode, median, and tistics, including the mode, median, and mean. mean.

There are three major types of estimatThere are three major types of estimates of central tendency:es of central tendency: MeanMean MedianMedian ModeMode

The The MeanMean or average is probably the most co or average is probably the most commonly used method of describing central temmonly used method of describing central tendency. To compute the mean all you do is adndency. To compute the mean all you do is add up all the values and divide by the number od up all the values and divide by the number of values. For example, the mean or average quf values. For example, the mean or average quiz score is determined by summing all the scoriz score is determined by summing all the scores and dividing by the number of students takies and dividing by the number of students taking the exam. For example, consider the test scng the exam. For example, consider the test score values:ore values:

15, 20, 21, 20, 36, 15, 25, 1515, 20, 21, 20, 36, 15, 25, 15 The sum of these 8 values is 167, so the mean iThe sum of these 8 values is 167, so the mean i

s 167/8 = 20.875.s 167/8 = 20.875.

The The MedianMedian is the score found at the exact mi is the score found at the exact middle of the set of values. One way to compute tddle of the set of values. One way to compute the median is to list all scores in numerical ordhe median is to list all scores in numerical order, and then locate the score in the center of ter, and then locate the score in the center of the sample. For example, if there are 500 scorehe sample. For example, if there are 500 scores in the list, score #250 would be the median. Is in the list, score #250 would be the median. If we order the 8 scores shown above, we woulf we order the 8 scores shown above, we would get:d get:

15,15,15,20,20,21,25,3615,15,15,20,20,21,25,36 There are 8 scores and score #4 and #5 represeThere are 8 scores and score #4 and #5 represe

nt the halfway point. Since both of these scorent the halfway point. Since both of these scores are 20, the median is 20. If the two middle scs are 20, the median is 20. If the two middle scores had different values, you would have to iores had different values, you would have to interpolate to determine the median.nterpolate to determine the median.

15,15,15,20,20,21,25,3615,15,15,20,20,21,25,36 The The modemode is the most frequently occurring is the most frequently occurring

value in the set of scores. To determine the value in the set of scores. To determine the mode, you might again order the scores as smode, you might again order the scores as shown above, and then count each one. The hown above, and then count each one. The most frequently occurring value is the mode.most frequently occurring value is the mode. In our example, the value 15 occurs three ti In our example, the value 15 occurs three times and is the model. In some distributions mes and is the model. In some distributions there is more than one modal value. For insthere is more than one modal value. For instance, in a bimodal distribution there are twtance, in a bimodal distribution there are two values that occur most frequently.o values that occur most frequently.

Notice that for the same set of 8 scores we gNotice that for the same set of 8 scores we got three different values -- 20.875, 20, and 15 ot three different values -- 20.875, 20, and 15 -- for the mean, median and mode respectiv-- for the mean, median and mode respectively. If the distribution is truly normal (i.e., bely. If the distribution is truly normal (i.e., bell-shaped), the mean, median and mode arell-shaped), the mean, median and mode are all equal to each other.e all equal to each other.

Dispersion.Dispersion. Dispersion refers to the sp Dispersion refers to the spread of the values around the central tread of the values around the central tendency. There are two common measendency. There are two common measures of dispersion, the range and the stures of dispersion, the range and the standard deviation. The andard deviation. The rangerange is simply is simply the highest value minus the lowest valthe highest value minus the lowest value. In our example distribution, the hiue. In our example distribution, the high value is 36 and the low is 15, so the gh value is 36 and the low is 15, so the range is 36 - 15 = 21. range is 36 - 15 = 21.

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Standard DeviationStandard Deviation