DESCRIPTIVE STATISTICS

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1 DESCRIPTIVE STATISTICS REVIEW OF KEY CONCEPTS SECTION. Measures of Locatio.. Arithmetic Mea xi x i x+ x + + x Cosider the data i Table.. They represet serum-cholesterol levels from a group of hospital workers who were regularly eatig a stadard U.S. diet ad who agreed to chage their diet to a vegetaria diet for a 6-week period. Cholesterol levels were measured before ad after adoptig the diet. The mea serum-cholesterol level before adoptig the diet is computed as follows: 4 x i i , x mgdl 4 Advatages. It is represetative of all the poits.. If the uderlyig distributio is Gaussia (bell shaped), the it is the most efficiet estimator of the middle of the distributio. 3. May statistical tests are based o the arithmetic mea. Disadvatages. It is sesitive to outliers, particularly i small samples; e.g., if oe of the cholesterol values were 800 rather tha 00, the the mea would be icreased by 5 mg/dl.. It is iappropriate if the uderlyig distributio is far from beig Gaussia; for example, serum triglycerides have a distributio that looks highly skewed (i.e., asymmetric).

2 STUDY GUIDE/FUNDAMENTALS OF BIOSTATISTICS 3 Table. Serum-Cholesterol levels before ad after adoptig a vegetaria diet (mg/dl) Subject Before After Before After Mea sd Alteratives to the Arithmetic Mea-Media Oe iterestig property from the table is that the diet appears to work best i people with high baselie levels versus people with low baselie levels. How ca we test if this is true? Divide the group i half, ad look at cholesterol chage i each half. To do this we must compute the media 50% poit i the distributio. Specifically, For example, F H..3 Stem-ad-Leaf Plots I K + Media th largest poit if is odd average of L th + F + th largest poits if is eve H I K O NM if 7, the the media 4th largest poit if 4, the the media average of (th + 3th) largest poit How ca we easily compute the media? We would have to order the data to obtai the th ad 3th largest poits. A easier way is to compute a stem-ad-leaf plot. Divide each data value ito a leaf (the least-sigificat digit or digits) ad a stem (the most-sigificat digit or digits) ad collect all data poits with the same stem o a sigle row. For example, the umber 95 has a stem of 9 ad a leaf of 5. A stem ad leaf plot of the before measuremets is give below. QP

3 4 CHAPTER /DESCRIPTIVE STATISTICS Cumulative total Stems Leaves Media 79 mg dl We have added a cumulative total colum which gives the total umber of poits with a stem that is the stem i that row. It is easy to compute the media from the stem ad leaf plot sice the media average of th ad 3th largest values ( ) 79. Note that the leaves withi a give row (stem) are ot ecessarily i order. Oe use of stem ad leaf plots is to provide a visual compariso of the values i differet data sets. The stem-ad-leaf plots of the chage i cholesterol for the subgroups of people below ad above the media are give as follows: 79 mg dl 80 mg dl The chage scores i the subgroups look quite differet; the subgroup with iitial value above the media is showig more chage. We will be able to test if the average chage score is sigificatly differet based o a t test (to be covered i Chapter 8 of the text)...4 Percetiles We ca also use stem-ad-leaf plots to obtai percetiles of the distributio. To compute the p th percetile, if p 0 is a iteger, the average the F H I K p p 0 th th largest poits Otherwise, pth percetile { p 0 + } the largest poit, where p 0 largest iteger p 0. For example, to compute the th percetile of the baselie cholesterol distributio, also kow as the lower decile, we have 4, p, p 0. 4, p 0, lower decile 3rd largest poit 5 mg/dl. To compute the 90th percetile (or upper decile), 4, p 90, p 0 6., p 0. Upper decile d largest poit 38 mg/dl.

4 STUDY GUIDE/FUNDAMENTALS OF BIOSTATISTICS 5 Media Commoly used percetiles, 0,,90% (deciles) 5, 50, 75% (quartiles) 0, 40,, 80% (quitiles) 33.3, 66.7% (tertiles) Advatages. Always guaratees that 50% of the data values are o either side of the media.. Isesitive to outliers (extreme values). If oe of the cholesterol values icreased from 00 to 800, the media would remai at 79 but the mea would icrease from 88 mg/dl to mg/dl. Disadvatages. It is ot as efficiet a estimator of the middle as the mea if the distributio really is Gaussia i that it is mostly sesitive to the middle of the distributio.. Most statistical procedures are based o the mea. We ca get a impressio of how symmetric a distributio is by lookig at the stem ad leaf plot. If we look at the stem ad leaf plot of the baselie values o the previous page we see that the distributio is oly slightly skewed, ad the mea may be adequate...5 Geometric Mea Oe way to get aroud the disadvatages of the arithmetic mea are to trasform the data oto a differet scale to make the distributio more symmetric ad compute the arithmetic mea o the ew scale. The most popular such scale is the l (atural log or log e ) scale: a f l x,, l ax f We ca ow take a average i the l scale ad deote it by l x : a f l x l x a f + + l x The problem with this is that the average is i the l scale rather tha the origial scale. Thus, we take the atilog of l x to obtai GM el x geometric mea The ERG (electroretiogram) amplitude (µ V ) is a measure of electrical activity of the retia ad is used to moitor retial fuctio i patiets with retiitis pigmetosa, a ofte-blidig ocular coditio. The followig data were obtaied from patiets to moitor the course of the coditio over a -year period. Year ERG amplitude (µ V ) Year ERG amplitude (µ V ) Absolute chage (µ V )

5 6 CHAPTER /DESCRIPTIVE STATISTICS The distributio of values at each year is highly skewed, with chage scores domiated by people with high year- ERG amplitudes. The distributio i the l scale is much more symmetric. Let s compute the GM for year ad year. Year l ( 9. ) + + l ( 63. ) l x 844. GM e µ V We ca quatify the % chage by Year l ( 4. ) + + l ( 35. ) l x GM e µ V GM GM % declie 0% ( ( )) Thus, the ERG has declied, o average, by 3.% over year. Geometric Mea Advatages. Useful for certai types of skewed distributios.. Stadard statistical procedures ca be used o the log scale. Disadvatages. Not appropriate for symmetric data.. More sesitive to outliers tha the media but less so tha the mea. SECTION. Measures of Spread.. Rage The rage the iterval from the smallest value to the largest value. This gives a quick feelig for the overall spread but is misleadig because it is solely iflueced by the most extreme values; e.g., cholesterol data iitial readigs; rage (37, 50)... Quasi-Rage A quasi-rage is similar to the rage but is derived after excludig a specified percetage of the sample at each ed; e.g., the iterval from the th percetile to the 90th percetile. For example, for the cholesterol data % poit 3rd largest from bottom 5 mg/dl 90% poit 3rd largest from top 38 mg/dl quasi-rage (5,38)

6 STUDY GUIDE/FUNDAMENTALS OF BIOSTATISTICS 7..3 Stadard Deviatio, Variace If the distributio is ormal or ear ormal, the the stadard deviatio is more frequetly used as a measure of spread. Why s rather tha s? x x s i sample variace i s sample stadard deviatio variace We wat a estimator of spread i the same uits as x ; i.e., if uits chage by a factor of c, ad the trasformed data is referred to as y, the y cx s cs but s c s y x a f y x Note that s chages by a factor of c (the same as x ), but s chages by a factor of c. Thus, s ad x ca be directly related to each other while s ad x caot. How ca we use x ad s to get a impressio of the spread of the distributio? If the distributio is ormal, the x ± s comprises about /3 of the distributio x ± s (more precisely,.96s) comprises about 95% of the distributio x ± 5. s (more precisely,.576s) comprises about 99% of the distributio If the distributio is ot ormal or ear ormal, the the distributio is ot well characterized by x, s. It is better to use the percetiles i this case (e.g., the media could be used istead of the mea ad the quasi-rage istead of the stadard deviatio). For example, for the cholesterol data, the variace ad stadard deviatio of the before measuremets are computed as follows: Let s see how ormal the distributio looks. 4 axi xf i s 5, s 33. x ± 96. s ± 96. ( 33. ) (. 8, 5. 8) icludes all poits; it should iclude 95% (or 3 out of 4 poits) uder a ormal distributio. x ± s ± 33. ( 54. 6,. ) icludes %. of poits; it should be /3 uder a ormal distributio. The ormal distributio appears to provide a reasoable approximatio. Note that computer programs such as Excel ca be used to compute may types of descriptive statistics. See the cd-rom for a example of usig Excel to easily compute the mea ad stadard deviatio. s

7 8 CHAPTER /DESCRIPTIVE STATISTICS..4 Coefficiet of Variatio (CV) CV s 0% x The CV is used if the variability is thought to be related to the mea. For the cholesterol data, CV 0% % SECTION.3 Some Other Meas for Describig Data.3. Frequecy Distributio This is a listig of each value ad how frequetly it occurs (or i additio, the % of scores associated with each value). This ca be doe either based o the origial values, or i grouped form; e.g., if we group the cholesterol chage scores by -mg icremets, the we would have Frequecy % 40.0, , < , < , < , < , < , < This ca be doe either i umeric or graphic form. If i graphic form, it is ofte represeted as a bar graph..3. Box Plot Aother graphical techique for displayig data ofte used i computer packages is provided by a Box plot. The box (rectagle) displays the upper ad lower quartiles, the media, arithmetic mea, ad outlyig values (if ay). It is a cocise way to look at the symmetry ad rage of a distributio. Outlyig value O Upper quartile Arithmetic mea Media Lower quartile O

8 STUDY GUIDE/FUNDAMENTALS OF BIOSTATISTICS 9 PROBLEMS... Suppose the origi for a data set is chaged by addig a costat to each observatio.. What is the effect o the media?. What is the effect o the mode?.3 What is the effect o the geometric mea?.4 What is the effect o the rage? Real Disease For a study of kidey disease, the followig measuremets were made o a sample of wome workig i several factories i Switzerlad. They represet cocetratios of bacteria i a stadard-size urie specime. High cocetratios of these bacteria may idicate possible kidey pathology. The data are preseted i Table.. Table. Cocetratio of bacteria i the urie i a sample of female factory workers i Switzerlad Cocetratio Frequecy Compute the arithmetic mea for this sample..6 Compute the geometric mea for this sample..7 Which do you thik is a more appropriate measure of locatio? Cardiovascular Disease The mortality rates from heart disease (per 0,000 populatio) for each of the 50 states ad the District of Columbia i 973 are give i descedig order i Table.3 []. Cosider this data set as a sample of size 5 ax, x,, x5f. 5 x i i 5 a i f i If 7, 409 ad x x 49, the do the followig: Table.3 Mortality rates from heart disease (per 0,000 populatio) for the 50 states ad the District of Columbia i 973 West Virgiia Wiscosi DC 37. Pesylvaia Vermot South Carolia Maie Nebraska Motaa Missouri 4.9 Teessee Marylad Illiois 40.8 New Hampshire Georgia Florida Idiaa Virgiia 3. 7 Rhode Islad North Dakota Califoria Ketucky Delaware Wyomig New York Mississippi Texas Iowa Louisiaa Idaho 97.4 Arkasas Coecticut Colorado 74.6 New Jersey Orego Arizoa Massachusetts Washigto Nevada Kasas Miesota Utah 4. 5 Oklahoma Michiga New Mexico Ohio Alabama Hawaii South Dakota North Carolia Alaska 83.9

9 CHAPTER /DESCRIPTIVE STATISTICS.8 Compute the arithmetic mea of this sample..9 Compute the media of this sample.. Compute the stadard deviatio of this sample.. The atioal mortality rate for heart disease i 973 was per 0,000. Why does this figure ot correspod to your aswer for Problem.8?. Does the differetial i raw rates betwee Florida (47.4) ad Georgia (3.8) actually imply that the risk of dyig from heart disease is greater i Florida tha i Georgia? Why or why ot? Nutritio Table.4 shows the distributio of dietary vitami-a itake as reported by 4 studets who filled out a dietary questioaire i class. The total itake is a combiatio of itake from idividual food items ad from vitami pills. The uits are i IU/0 (Iteratioal Uits/0). Table.4 Distributio of dietary vitami-a itake as reported by 4 studets Studet umber Itake (IU/0) Studet umber Itake (IU/0) Compute the mea ad media from these data..4 Compute the stadard deviatio ad coefficiet of variatio from these data..5 Suppose the data are expressed i IU rather tha IU/0. What are the mea, stadard deviatio, ad coefficiet of variatio i the ew uits?.6 Costruct a stem-ad-leaf plot of the data o some coveiet scale..7 Do you thik the mea or media is a more appropriate measure of locatio for this data set? SOLUTIONS.... Each data value is chaged from x i to xi + a, for some costat a. The media also icreases by a.. The mode icreases by a..3 The geometric mea is chaged by a udetermied amout, because the geometric mea is give by atilog laxi + af ad there is o simple relatioship betwee l a xi + a f ad l af. x i.4 The rage is ot chaged, sice it is the distace betwee the largest ad smallest values, ad distaces betwee poits will ot be chaged by shiftig the origi..5 The arithmetic mea is give by ( ) ( ) 7.6 To compute the geometric mea, we first compute the mea log to the base as follows: a f a f 5 log log The geometric mea is the give by The geometric mea is more appropriate because the distributio is i powers of ad is very skewed. I the log scale, the distributio becomes less skewed, ad the mea provides a more cetral measure of locatio. Notice that oly 33 of the 77 data poits are greater tha the arithmetic mea, while 46 of the 77 poits are greater tha the geometric mea..8 We have that x 7, per 0, Sice 5 is odd, the media is give by the ( 5 + ) th or 6th largest value mortality rate for Mississippi 35.6 per 0,000.

10 STUDY GUIDE/FUNDAMENTALS OF BIOSTATISTICS. We have that ( x ) i x i 49, , s Thus, s 4, per 0,000. The atioal mortality rate is a weighted average of the state-specific mortality rates, where the weights are the umber of people i each state. The arithmetic mea i Problem.8 is a uweighted average of the state-specific mortality rates that weights the large ad small states equally.. No. The demographic characteristics of the residets of Florida may be very differet from those of Georgia, which would accout for the differece i the rates. I particular, Florida has a large retiree populatio, which would lead to higher mortality rates. I order to make a accurate compariso betwee the states, we would, at a miimum, eed to compare disease rates amog specific age-sex-race groups i the two states x Media average of the 7th ad 8th largest values s 4 a xi x i 3 358, s s CV 0% x 0% f 67. 9%.5 Mea , 687 IU, s , IU, CV 67. 9% (uchaged)..6 We will roud each umber to the earest iteger i costructig the stem-ad-leaf plot: The media is more appropriate, because the distributio appears to be skewed to the right. REFERENCE... [] Natioal Ceter for Health Statistics. (975, February ), Mothly vital statistics report, summary report, fial mortality statistics (973), 3() (Suppl. ).

Chapter 2 Descriptive Statistics

Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data