Statisticalmeasures are useful in presenting any information or data in ameaningful way. Each statistical test has some salient features whichdetermine its appropriateness in statistical analysis.
Measures of central tendency
Theyare also known as measures of central location and they give a roughidea of the position of the point around which other observationscluster. There are three primary measures of the central locationwhich include the mean, mode, and median.
Themean: Itcan be further categorized into the arithmetic mean, geometric meanand the harmonic mean (Dugan, 2012).
The arithmetic mean
Itis defined as the sum of all items in the sample or the populationdivided by the sample or population size respectively. We can haveungrouped and grouped data.
Ungroupeddata:where n is the sample size and are the individual observations.
Groupeddata:whereis the frequency and are the different observations
The Geometric Mean (G.M)
Itis the root of the product of all elements in a sample or the population. Itis mainly applicable when the data were given in the form of ratios.
The Harmonic Mean (H.M)
Itis determined by dividing the sample size (with the sum of the reciprocals of the individual observationsappearing in the sample. It is applicable when the data has manyoutliers.
Ungroupeddata: and grouped data:
Themedian: Themedian is simply the middle value in a given data set after theobservations have been arranged in a descending or ascending order(Dugan,2012).
Ungroupeddata: whenthe sample size (n) is an odd number or whenthe sample size (n) is an even number.
Groupeddata: whereis the lower class limit of the median class, is the frequency of the median class, is the cumulative frequency of the median class and is the sample size.
Themode: Itis the most occurring observation in a given data set. For ungroupeddata, there is no formula for calculating the modal value we justneed to observe the frequently repeated value and quote it as themode. However, in the grouped data we can obtain the modal valueusing the following formula.
However,the above formula is only applicable if the class sizes are equal. Insome cases, we may encounter situations where the class widths areunequal. When the class sizes are unequal, we do not use frequenciesto calculate the modal value. We calculate the height of the classinterval by dividing the frequency by that class width. We thenreplace the frequencies with the heights and apply the formula asstated above
Measures of variability
Measuresof variability or dispersion are useful in determining the spread orthe variation of the individual observations from the mean. Thevariance and the standard deviation are the most known measures ofspread. Others include the range and the interquartile range.
Range:It is the basic measure of dispersion or spread, and it can becalculated by obtaining the difference between the highest and theleast value in the sample (Cowan, 2011).
InterquartileRange(IR): Itcan be calculated by getting the difference between the 75thpercentile (Q3)and the 25thpercentile (Q1).Thereforewhere:
andWherethelower class boundary, isthe frequency, isthe cumulative frequency, is the sample size and is the class size.
Varianceand standard deviation: Varianceis the degree of variation of the individual observations from theirmean denoted bywhile the standard deviation is the square root of the variancedenoted by. and
Measures of position
Theyfocus on determining the position of a given observation relative toother observations in a given sample or population. The primarymeasures of position include the percentiles, quartiles and thestandard scores also known as the Z- scores (Anderson& Finn, 2013).
Percentilesand Quartiles: Assumingthat the data has already been ordered in an ascending manner,percentiles divide the rank-ordered elements equally into 100 parts.On the other hand, quartiles divide the sample observations equallyinto 4 parts. This implies that we have the 1st,2ndand the 3rdquartiles abbreviated as respectively. The percentiles are denoted by . The 50thpercentile ()is equal to the 2ndquartile (and they correspond to the median value of any data set. On similarlines, the first quartile (Q1)corresponds with the 25thpercentile ()and the 3rdquartile (Q3)corresponds with the 75thpercentile (P75).The formula for obtaining the percentiles and the quartiles aresimilar. Assume that we want to get the 50thpercentile or the 2ndquartile. We would have:
andWherethelower class limit, isthe frequency, isthe cumulative frequency, is the sample size and is the class size.
Thestandard score (Z-score): Itestimates how many standard deviations an observation is from themean. where is the sample element, is the population mean and isthe standard deviation.
The five-numbers summary
Itis a descriptive statistics that give information about a set ofsample elements. This information entails the five most criticalsample statistics. They include the sample maximum value ( largestelement), the sample minimum value (smallest observation), the medianvalue which is also known as the 50thpercentile or the 2ndquartile, the lower quartile ( 25thpercentile) and the upper quartile or the 3rdquartile also referred to as the 75thpercentile (Cowan,2011).The formula to obtaining these percentiles does not change. It is thesame as the one discussed under measures of position (Anderson& Finn, 2013).
Anderson,T. & Finn, J. (2013). Thenew statistical analysis of data.New York: Springer.
Cowan,G. (2011). Statisticaldata analysis.Oxford: Clarendon Press.
Dugan,C. (2012). Tornadochasers:Measures of Central Tendency.Huntington Beach, CA: Teacher Created Materials.