K. Leif. Lincoln University, San Francisco California.
The Those with one or more siblings totally deaf discount top avana 80mg otc, but hearing factors responsible for such results are not immediately clear parents; and certainly more research is needed in this ﬁeld buy 80mg top avana. Those with one or more siblings with some hearing difﬁ- culties purchase top avana 80 mg without prescription, but hearing parents; Those with neither parents nor siblings with hearing problems buy 80 mg top avana. Effects of a family history The ﬁrst four groups were each compared with group 5 after of hearing problems in adults controlling for the demographic and other variables considered in the earlier analysis. The Blue differed from those with hearing parents Mountain survey combined audiometry and questionnaires and was administered to 2956 participants aged 49 years and older. These indicated that, after controlling for age a Engagement in education Better and sex, those with a parental family history of hearing loss had sig- Quality of life niﬁcantly worse hearing than those without (Fig. This shows that while the survey, the question “Do you have difﬁculty with your hearing? In the the same in the two groups, the mean hearing level for those Blue Mountain Survey, the question “Do you feel you have a reporting no hearing difﬁculties is lower in the group with hearing loss? It may be seen from this ﬁgure that the proportions the results of the generic question with greater difﬁculties with a family history reporting difﬁculties were almost identical reported by those with a family history. This could be related to the fact that older of the level of hearing difﬁculties when we considered other subjects, on the whole, complain of hearing problems only when effects of having a family history. Elsewhere, in a group of patients with tinnitus, the rela- The question then arises as to how much of this family tionship between these “surrogate” measures and the hearing history effect relates to the differences in the hearing thresh- levels has been examined (27). We was considered was the annoyance caused by the hearing difﬁ- culty “Nowadays how much does any difﬁculty in hearing worry, annoy or upset you? Similarly increased levels of 40 annoyance in the presence of a family history were also found for those with slight difﬁculties hearing the television and are % 30 presented elsewhere (23). These results are shown in These show again that, after controlling for the level of Figure 10. This indicates that for all levels of hearing difﬁculty, reported hearing difﬁculties, those individuals with a family his- those with a family history of hearing loss are more likely to tory of hearing impairment are more annoyed by loud sounds report tinnitus than those without such a family history. The annoyance caused by the tinnitus and the effects of the The other aural symptom to be considered in these analy- tinnitus on the individual’s life was also examined in both stud- ses was tinnitus. In the Blue Mountain study, no signiﬁcant effects were get noises in your head or ears (tinnitus), which usually last found in this respect. While the questions in from the two studies are not clear and could well be due to the the two studies regarding tinnitus differed, both indicated that different criteria for the family history, as well as from different tinnitus was found more commonly in individuals with a family wording of the questions in the two studies. The equivalent ﬁgures from the Blue Mountain as a function of whether the tinnitus is present some or most of survey were 35. It may be seen that having such a family history nitus (28) so, to control for this, the reported tinnitus in the results in greater annoyance provoked by the tinnitus. The only signiﬁcant differences found in the work situ- ation and work experience were that females were more likely to be in work that had required a university education if they 2 had a family history of hearing problems ( 8. This is despite having generally poorer hearing levels than those without such a family history. These results No Annoyance Slight Moderate Severe are broadly in line with the previous study (29) and a number Annoyance Annoyance Annoyance of the studies discussed earlier in the section of the inﬂuence of Level of annoyance having such a family history in children. Greater effects in the presence of a family history of three different approaches will be described. Firstly, the effects hearing impairment, even when controlled for hearing level of simply asking patients seen in a clinic, or subjects contacted and tinnitus occurrence, were found (23). Most interestingly, via the internet, to list the effects on them of having such a even when controlling for the degree of annoyance evoked by family history (31,32) will be described. Secondly, the results the tinnitus, if the subject had a family history of hearing loss, obtained using a structured questionnaire based on the results of the reported effect on their life was greater (Fig. Finally, the results of ever, this effect was most pronounced at the lower levels of some in-depth interviews of patients with a family history of annoyance. In the ﬁrst of our open-ended studies (31), patients attend- ing audiological rehabilitation clinics who were found to have Family history effects in working age adults a family history of hearing impairment during the clinical inter- In addition to the broad retrospective population studies view were administered the following questionnaire: approached by secondary analysis, two studies have been per- “You have mentioned that other members of your family have or formed in Sweden on working age adults in which the experi- have previously had hearing problems. Does this information have menters returned to a previous experimental group with any effect on your reaction to your own hearing problems? The second comprised 445 The study had three speciﬁc aims: respondents with hearing impairment, of onset at a variety of ■ To determine what proportion of such patients saw their ages, seen in various centres in Sweden, and who were still in family history as having an effect on them the workforce. The individuals seen in audiological rehabilitation clinics and who only signiﬁcant differences between the two groups were that had a family history of hearing difﬁculties. Two-thirds were those with a family history were more likely to have had a uni- female and they had a median age of 67 years. These responded with yes to the question “Do you want to Among these respondents, a total of 150 “effects” were listed. The ﬁrst category of occurred despite the fact that the two groups did not differ “general effects” covers those unrelated to the family history, signiﬁcantly across a range of demographic variables. The “positive” and “negative” categories are self-explanatory 45 years in the respondent’s mother and/or father and/or onset and will be considered further below. The “neutral” category before the age of 20 years in the respondent’s brother and/or comprised those responses, which did not obviously entail Psychosocial aspects of genetic hearing impairment 151 Table 10. From this it may be seen that role modelling, help-seeking, and Positive effects 68 sharing knowledge are predominantly characterized by positive Negative effects 23 reactions. Acceptance and “worry about the future/offspring” “Neutral” effects 29 evoke predominantly negative reactions, and expectation/ anticipation evokes a largely neutral response. Based on the most commonly found responses from these open-ended questionnaires, which indicated an effect of having either a positive or a negative effect on the respondent (e. The questionnaire was adminis- providing a better understanding of their own and others’ prob- tered to groups of patients in Cardiff who indicated that they lems. These, as well as the negative responses will be considered had a family history of hearing impairment, and also to those further below. Such negative responses were centred around subjects who had responded by internet to the open-ended concerns for their own future or for that of their children questionnaire in the previous study (32). For the last, that 18 of the 20 items related to most of the others, the excep- the questions were translated into Dutch (32). Almost all tions being item 4 (“I didn’t realize hearing problems were heredi- the responses came from the Dutch website, and only one tary”) and item 11 (“I am not worried about using hearing aids, as respondent out of 41 indicated that the family history had I know how much of a problem it is for others without one”). In all, 90 speciﬁc responses were obtained, almost therefore excluded these two items from a factor analysis, which equally divided between the “positive,” “negative,” and “neu- subsequently identiﬁed ﬁve factors, accounting for 58. Of these, two factors had acceptable In this study, the main aim, apart from a comparison with coefﬁcients and are shown in Table 10. This entailed negative effects of the family history (three items – factor 2, deriving “themes” from the “meaning units” or responses. Factor 2 was Positive 10 Negative signiﬁcantly related to overall hearing level (p 0. The former relationship indicates Meaning units that the more severe the experienced hearing loss, the more 6 (n) negative the respondents consider the impact of a family history 4 to be. A factor analysis on this group of questions revealed three factors accounting for 56. Within these, the two Cardiff groups questionnaire with factor loadings, total variance and generally gave the same response, but differed from the website coefﬁcients of the two main factors group who were also younger. Again one of the most isolation important factors to emerge is whether the individual had 3 Family history – – 0. In both groups, there is problem some effect on transactional communication, but in the aware 7 Open about – – 0. And also the teasing to seek help “you’re getting deaf now and that kind of thing, so I was encouraged sooner by the family. They consid- in life because of family history ered adults who were predominantly late middle aged and, in general, only very minor differences were found between those 15 Knowledge about – – 0. They were children’s future hearing problems taking part in an aetiological and genetic study on age-related hearing impairment. Fifty-one had no family history of hearing Psychosocial aspects of genetic hearing impairment 153 impairment and 58 did. Their mean better ear hearing level was The second study (38) had two components, a secondary 38. There were no signiﬁcant analysis of an earlier study, which had looked the effects of differences in gender, age, or hearing level between the two motivation on hearing aid outcome measures (43) and a groups. In the former, case ﬁles on 58 using the quantitative Denver Scale (40), and depression and patients, attending a clinic to obtain hearing aids for the ﬁrst anxiety were assessed using the Hospital Anxiety and Depression time, were reviewed to obtain details of whether or not they Scale (41). Overall scores for both scales showed no signiﬁcant had a family history of hearing problems. Thirty-one had such difference between the two groups of subjects, although some a family history and 27 did not.
The Y intercept indicates the starting point from which the Y scores begin to change buy discount top avana 80 mg online. Thus cheap 80 mg top avana otc, together buy top avana 80mg free shipping, the slope and intercept describe how effective 80mg top avana, starting at a particular Y score, the Y scores tend to change by a specific amount as the X scores increase. As an example, say that we have developed a test to identify (predict) those indi- viduals who will be good or bad workers at a factory that makes “widgets. The predictor (X) variable is participants’ scores on the widget test, and the criterion (Y) variable is the number of widgets they produced. This is a very strong, positive linear relationship, and so the test will be what researchers call “a good predictor” of widget-making. The numerator of the formula for b is the same as the numerator in the formula for r, and the denominator of the formula for b is the left-hand quantity in the denominator of the formula for r. This positive slope indicates a positive relationship, which fits with the positive r of 1. Had the rela- tionship been negative, the formula would have produced a negative number here. Computing the Y Intercept The formula for the Y intercept of the linear regression line is a 5 Y 2 1b21X2 First, multiply the mean of all X scores times the slope of the regression line. Describing the Linear Regression Equation Once you have computed the Y intercept and the slope, rewrite the regression equation, substituting the computed values for a and b. Plotting the Regression Line We use the finished regression equation to plot our linear regression line. To draw a line, we need at least two data points, so choose a low and high X score, insert each into the regression equation, and compute the Y¿ for that X. Therefore, we also use the finished regression equation to predict anyone’s Y score if we know their X score. In fact, computing any Y¿ using the equation is the equivalent of going to the graph and traveling vertically from the X score up to the regression line and then left to the value of Y¿ on the Y axis. We can compute Y¿ for any value of X that falls within the range of Xs in our data, even if it’s a score not found in the original sample: No one scored an X of 1. Our regression equa- tion is based only on widget test scores between 1 and 4, so we shouldn’t predict a Y for an X of, for example, 6. This is because we can’t be sure what the nature of the relationship is at 6—maybe it’s curvilinear or has a steeper slope. Putting all of this together, the preceding computations are summarized in Table 8. Substitute the values of a and b into the formula for the regression equation: Y¿ 5 1b21X2 1 a 5. The components of the regression equation to To use X to predict Y in these scores, compute first are the ______ and ______. Compute b for the following scores: X Y X Y 1 1 2 2 3 3 4 Compute b: ©X 5 12, ©Y 5 25, ©X2 5 28, 4. To describe the amount of prediction error we expect when predicting unknown scores, we first determine how well we can predict the actual Y scores in our sample: We pretend we don’t know the scores, predict them, and then compare the predicted Y¿ scores to the actual Y scores. The error in a single prediction is the amount that a participant’s Y score differs from the corresponding predicted Y¿ score: In symbols this is Y 2 Y¿, and it is literally the dif- ference between the score a participant got and the score we predict he or she got. The predictions for some participants will be closer to their actual Y scores than for others, so we would like to compute something like the average error across all predictions. To find the average error, we first compute Y¿ for everyone in the sample and sub- tract their Y¿ from their actual Y score. Statisticians equate errors with deviations, so Describing Errors in Prediction 169 Y 2 Y¿ equals the amount that Y deviates from Y¿. To get the average error, we would like to simply sum these deviations and then find the average, but we cannot. Therefore, the Ys are equally spread out around their Y¿ scores, in the same way that previously we saw that Xs are spread out around their X. Because of this, like with the mean, the positive and nega- tive deviations with Y will cancel out, always producing a sum equal to zero. The sum of the squared deviations of Y 2 Y¿ is not necessarily zero, so neither is the average squared deviation. Computing the Variance of the Y Scores Around Y9 The variance of the Y scores around Y¿ is the average squared difference between the actual Y scores and their corresponding predicted Y¿ scores. The S2 indicates sample variance or error, and the subscript Y¿ indi- Y¿ cates that it is the error associated with using Y¿ to predict Y scores. The formula that defines the variance of the Y scores around Y¿ is ©1Y 2 Y¿ 22 S2 5 Y¿ N Like other definitional formulas we’ve seen, this formula is important because it shows the core calculation involved: We subtract the Y¿ predicted for each participant from his or her actual Y score giving us a measure of our error. The answer is one way to measure roughly the “average” amount of error we have when we use linear regression to predict Y scores. Note: Among the approaches we might use, the regression procedures described in this chapter produce the smallest error in predictions possible, thereby producing the smallest sum of squared deviations possible. In the defining formula, we can replace Y¿ with the formulas for finding Y¿ (for finding a, b, and so on). Among all of these formulas we’ll find the com- ponents for the following computational formula. The computational formula for the variance of the Y scores around Y9 is S2 5 S2 11 2 r22 Y¿ Y Much better! Therefore, finish the computations of S2 using the formula at the begin- Y ning of this chapter. Although this variance is a legitimate way to compute the error in our predictions, it is only somewhat like the “average” error, because of the usual problems when interpreting variance. First, squaring each difference between Y and Y¿ produces an unrealistically large number, inflating our error. Second, squaring produces error that is measured in squared units, so our predictions above are off by 2. To distinguish the standard deviation found in regression, we call it the standard error of the estimate. Computing the Standard Error of the Estimate The standard error of the estimate is similar to a standard deviation of the Y scores around their Y¿ scores. It is the clearest way to describe the “average error” when using Y¿ to predict Y scores. By computing the square root, the answer is a more realistic number and we are no longer dealing with a squared variable. The core calcu- lation, however, is still to find the error between participants’ actual Y scores and their predicted Y¿ scores, and this is as close as we will come to computing the “average error” in our predictions. Then we find the square root of the quantity 1 2 r2 and then multiply it times the standard deviation of all Y scores. Therefore, we conclude that when using the regression equation to predict the number of widgets produced per hour based on a per- son’s widget test score, when we are wrong, we will be wrong by an “average” of about 1. It is appropriate to compute the standard error of the estimate anytime you compute a correlation coefficient, even if you do not perform regression—it’s still important to know the average prediction error that your relationship would produce. The symbol for the variance of the Y scores around errors in prediction when using regression, which Y¿ is ______. Y¿ Y¿ Interpreting the Standard Error of the Estimate In order for S (and S 2) to accurately describe our prediction error, and for r to accu- Y¿ Y¿ rately describe the relationship, you should be able to assume that your data generally meet two requirements. Homoscedasticity occurs when the Y scores are spread out to the same degree at every X. Because the vertical spread of the Y scores is constant at every X, the strength of the relationship is relatively constant at both low Xs and at high Xs, so r will accurately describe the relationship for all Xs. Further, the vertical distance sepa- rating a data point above or below the regression line on the scatterplot is a way to visualize the difference between someone’s Y and the Y¿ we predict. Heteroscedasticity occurs when the spread in Y is not equal throughout the relationship. Now part of the relationship is very strong (forming a nar- row ellipse) while part is much weaker (forming a fat ellipse). Therefore, r will not accurately describe the strength of the relationship for all Xs. Second, we assume that the Y scores at each X form an approximately normal distri- bution. That is, if we constructed a frequency polygon of the Y scores at each X, we should have a normal distribution centered around Y¿. Recall that in a normal distribution approximately 68% of the scores fall between ;1 standard deviation from the mean.
Serious infections associated with anticytokine therapies in the rheumatic diseases cheap top avana 80 mg on-line. Life-threatening histoplasmosis complicating immunotherapy with tumor necrosis factor alpha antagonists infliximab and etanercept purchase 80mg top avana mastercard. Pneumonia due to Cryptococcus neoformans in a patient receiving infliximab: possible zoonotic transmission from a pet cockatiel cheap top avana 80mg with mastercard. Pulmonary cryptococcosis after initiation of anti-tumor necrosis factor-a therapy [letter] discount top avana 80mg with amex. Disseminated cryptococcal infection in rheumatoid arthritis treated with methotrexate and infliximab. Pneumocystis carinii pneumonia associated with low dose methotrexate treatment for rheumatoid arthritis. Pneumocystis jiroveci (carinii) pneumonia after infliximab therapy: a review of 84 cases. Absence of tumour necrosis factor facilitates primary and recurrent herpes simplex virus-1 infections. Perioperative management of patients with rheumatoid arthritis in the era of biologic response modifiers. The risk of post-operative complications associated with infliximab therapy for Crohn’s disease: a controlled cohort study. Infectious and healing complications after elective orthopaedic foot and ankle surgery during tumor necrosis factor–alpha inhibition therapy [abstr]. Risk factors for surgical site infections and other complications in elective surgery in patients with rheumatoid arthritis with special attention for anti- tumor necrosis factor: a large retrospective study. Tumor necrosis factor inhibitor therapy and risk of serious postoperative orthopedic infection in rheumatoid arthritis. Infections during tumour necrosis factor-a blocker therapy for rheumatic diseases in daily practice: a systematic retrospective study of 709 patients. Rates of serious infection, including site-specific and bacterial intracellular infection, in rheumatoid arthritis patients receiving anti-tumor necrosis factor therapy. Ledingham J, Deighton C, British Society for Rheumatology Standards, Guidelines and Audit Working Group. Thrombotic thrombocytopenic purpura and clopidogrel: a need for new approaches to drug safety. Adverse drug event reporting in intensive care units: a survey of current practices. In fact, infections are the most common indication for admissions of transplant recipients in emergency departments (35%), and severe sepsis (11. Antimetabolite immunosuppressive drugs such as mycophenolate mofetil and azathio- prine are associated with significantly lower maximum temperatures and leukocyte counts (10). However, in general, the immunosuppression caused by transplantation does not abolish the inflammatory response, so most transplant recipients with a significant infection will have fever and most fevers will have an infectious etiology in this setting. Accordingly, many of these patients will be cared by physicians not always familiar with the specific problems posed by the transplant population. Where no solid data were available, perspectives based on our own experience and opinion are presented. Infections are more frequent and severe than those occurring in renal transplant recipients, but less frequent than those occurring after a liver or a lung transplantation. Importance of the Underlying Disease and Type of Transplantation The type of organ transplanted, the degree of immunosuppression, the need for additional antirejection therapy, and the occurrence of technical or surgical complications, all impact on the incidence of infection posttransplant. In each type of transplantation, there are patients in which the risk of infection is greater. Incidence of infection is higher in thoracic transplantation pediatric population than that in adult (17). Thrombocytopenia of <50 Â 10 /L for three days is frequent after liver transplantation and as such was not found to be an important contributor to bleeding. If severely ill patients with end-stage liver disease are selected appropriately, liver transplant outcomes are similar to those observed among subjects who are less ill and are transplanted electively from home (20). Patients receiving alemtuzumab for the treatment of allograft rejection are more prone to suffer opportunistic infections (23,24). Infections such as insertion site sepsis, endocarditis, pneumonia, candidiasis, or sternal infection may complicate 38% of support courses. The use of extended donors does not seem to increase the risk of poor outcome (31). The time of appearance of infection after transplantation is an essential component of the evaluation of the etiology of infection. Early infections occurring in transplant patients within the first month after transplantation are generally similar to that in nontransplant patients who have undergone major surgery in the same body area. Reactivation of latent infections and early fungal and viral infections account for a smaller proportion of febrile episodes during this period. Finally, late infections (after 6 months) may be caused either by common community pathogens in healthy patients or by opportunistic microorganisms in patients with chronic rejection. Some of these may not be evident during the initial examination, which should be frequently repeated. If the patient is still intubated and the chest X ray does not reveal infiltrates, the possibility of tracheobronchitis or bacterial sinusitis should be considered. Herpetic stomatitis and infections transmitted with the allograft or present in the recipient may also appear at this time. Intermediate Period From the second to the sixth month, patients are susceptible to opportunistic pathogens that take advantage of the immunosuppressive therapy. In this period, we may expect infection with immunomodulatory viruses and with opportunistic pathogens (P. Some bacterial infections such as listeriosis may appear at this time as primary sepsis or meningitis. Aspergillosis may be encountered in patients with risk factors or massive exposure (39) and toxoplasmosis in seronegative recipients of a seropositive allograft (40). At this time, fever of unknown origin should be managed almost as in immunocompetent hosts. However, the aforementioned opportunistic infections may compli- cate this late period in patients with chronic viral infection such as hepatitis B or C, which may progress to end-stage organ dysfunction and/or cancer. Patients requiring chronic hemodialysis or with malignancy or late rejection are also susceptible to opportunistic infections (Cryptococcus neoformans, P. Previous infections or colonization, exposure to tuberculosis, contact with animals, raw food ingestion, gardening, prior antimicrobial therapy or prophylaxis, vaccines or immunosuppressors, and contact with contaminated environment or persons should be recorded (42,43). History of residence or travel to endemic areas of regional mycosis (44) or Strongyloides stercoralis may be essential to recognize these diseases (45). Exposure to ticks may be essential to diagnose entities such as human monocytic ehrlichiosis, which may be potentially lethal in immunosuppressed patients (46). Certain complications may increase the risk of bacterial and fungal infections in the early posttransplant period (Table 2). They include long operation (over 8 hours), blood transfusion in excess of 3 L, allograft dysfunction, pulmonary or neurological problems, diaphragmatic dysfunction, renal failure, hyperglycemia, poor nutritional state, and thrombocytopenia (18,47–50). Within the exploration of the thoracic area, the consultant should visualize the entry sites of all intravascular devices, even if they “have just been cleansed. Sepsis, without local signs, may be the initial sign of postsurgical mediastinitis. When the sternal wound remains closed, a positive epicardial pacer wire culture may be a clue to sternal osteomyelitis (55). Its presence requires rapid debridement and effective antimicrobial therapy and should prompt the exclusion of adjacent cavities or organ infection. If ascites is present, it should be immediately analyzed and properly cultured to exclude peritonitis. We recommend bedside inoculation in blood-culture bottles due to its higher yield of positive results. Tenderness, erythema, fluctuance, or increase in the allograft size may indicate the presence of a deep infection or rejection. Finally, skin and retinal examinations are “windows” at which the physician may look in and obtain quite useful information on the possible etiology of a previously unexplained febrile episode. We have analyzed the value of ocular lesions in the diagnosis and prognosis of patients with tuberculosis, bacteremia, and sepsis (59,60). Cutaneous or subcutaneous lesions are a valuable source of information and frequently allow a rapid diagnosis. Viral and fungal infections are the leading causes of skin lesions in this setting.
For example discount top avana 80mg overnight delivery, say that a group of students ranked how well a college professor teaches cheap top avana 80 mg amex. Reporting that the professor’s median ranking was 3 com- municates that 50% of the students rated the professor as number 1 cheap 80 mg top avana free shipping, 2 buy 80mg top avana free shipping, or 3. Also, as you’ll see later, the median is preferred when interval or ratio scores form a very skewed distribution. Computing the median still ignores some information in the data because it reflects only the frequency of scores in the lower 50% of the distribution, without considering their mathematical values or considering the scores in the upper 50%. Therefore, the median is not our first choice for describing the central tendency of normal distribu- tions of interval or ratio scores. Although technically we call this statistic the arithmetic mean, it is what most people call the average. Compute a mean in the same way that you compute an average: Add up all the scores and then divide by the number of scores you added. Usually, we use X to stand for the raw scores in a sample and then the symbol for a sample mean is X. It is pronounced “the sample mean” (not “bar X”: bar X sounds like the name of a ranch! Get in the habit of thinking of X as a quantity itself so that you understand statements such as “the size of X ” or “this X is larger than that X. Then The formula for computing a sample mean is ΣX X 5 N As an example, consider the scores 3, 4, 6, and 7. Saying that the mean of these scores is 5 indicates that the mathematical center of this distribution is located at the score of 5. Uses of the Mean Computing the mean is appropriate whenever getting the “average” of the scores makes sense. The mathematical center of the distribution must also be the point where most of the scores are located. For example, say that a simple creativity test produced the scores of 5, 6, 2, 1, 3, 4, 5, 4, 3, 7, and 4, which are shown in Figure 4. Computing the mean is appropriate here be- Location of the mean on a symmetrical distribution. Always compute the mean to summarize a 3 normal or approximately normal distribution: The mean is the mathematical center of any distribution, and in a 2 f normal distribution, most of the scores are located around this central point. Therefore, the mean is an accurate sum- 1 mary and provides an accurate address for the distribution. With normally distributed or symmetrical distributions To find the mean of the scores 3, 4, 6, 8, 7, 3, and 5: of interval or ratio scores ΣX 5 3 1 4 1 6 1 8 1 7 1 3 1 5 5 36, and N 5 7. Comparing the Mean, Median, and Mode In a perfect normal distribution, all three measures of central tendency are located at the same score. If a distribution is only roughly normal, then the mean, median, and mode will be close to the same score. In this case, you might think that any measure of central tendency would be good enough. The mean uses all information in the scores, and most of the inferential procedures we’ll see involve the mean. Therefore, the rule is that the mean is the preferred statistic to use with interval or ratio data unless it clearly provides an inaccurate summary of the distribution. You have seen this happen if you’ve ever obtained one low grade in a class after receiving many high grades—your average drops like a rock. What hurts is then telling someone your average because it’s mis- leading: It sounds as if all of your grades are relatively low, even though you have only that one zinger. The mean is always pulled toward the tail of any skewed distribution because it must balance the entire distribution. You can see this starting with the symmetrical distribu- tion containing the scores 1, 2, 2, 2, and 3. As this illustrates, although the mean is always at the mathematical center, in a skewed distribution, that center is not where most of the scores are located. In both graphs, the mean is pulled toward the extreme tail and is not where most scores are located. The mode is toward the side away from the extreme tail and so the distri- bution is not centered here either. Thus, of the three measures, the median most accu- rately reflects the central tendency—the overall address—of a skewed distribution. It is for the above reasons that the government uses the median to summarize such skewed distributions as yearly income or the price of houses. But a relatively small number of corporate executives, movie stars, professional athletes, and the like make millions! However, this is misleading because most people do not earn at or near this higher figure. In sum, the first step in summarizing data is to compute a measure of central tendency to describe the score around which the distribution tends to be located. High scores scores Deviations Around the Mean 69 Most often the data in behavioral research are summarized using the mean. This is because most often we measure variables using interval or ratio scores and most often with such scores, “mother nature” produces a reasonably normal distribution. Because the mean is used so extensively, we will delve further into its characteristics and uses in the following sections. The simplest transformation is to add, sub- tract, multiply, or divide each score by a constant. If we add a constant 1K2 to each raw score in a sample, the new mean of the trans- formed scores will equal the old mean of the raw scores plus K. In essence, using a con- stant merely changes the location of each score on the variable by K points, so we also move the “address” of the distribution by the same amount. The mean is the center score because it is just as far from the scores above it as it is from the scores below it. That is, the total distance that some scores lie above the mean equals the total distance that other scores lie below the mean. The distance separating a score from the mean is called the score’s deviation, indi- cating the amount the score “deviates” from the mean. A score’s deviation is equal to the score minus the mean, or in symbols: The formula for computing a score’s deviation is X – X Thus, if the sample mean is 47, a score of 50 deviates by 13, because 50 2 47 is 1 3. Thus, you can see that a posi- tive deviation indicates that the raw score is larger than the mean and graphed to the right of the mean. A negative deviation indicates that the score is less than the mean and graphed to the left of the mean. The size of the deviation (regardless of its sign) in- dicates the distance the raw score lies from the mean: the larger the deviation, the farther into the tail the score is from the mean. If we add together all of the positive deviations we have the total distance that some scores are above the mean. Adding all of the negative deviations, we have the total distance that other scores are below the mean. If we add all of the positive and negative deviations together, we have what is called the sum of the deviations around the mean. Thus, the mean is the center of a distribution because, in total, it is an equal distance form the scores above and below it. Therefore, the half of the distribution that is below the mean balances with the half of the distribution that is above the mean. The sum of the deviations around the mean always equals zero, regardless of the shape of the distribution. For example, in the skewed sample of 4, 5, 7 and 20, the mean is 9, which produces deviations of 25, 24, 22, and 111, respectively. Some of the formulas you will see in later chapters involve something similar to computing the sum of the deviations around the mean. The statistical code for finding the sum of the deviations around the mean is Σ1X 2 X2. The Σ indicates to then find the sum The mean is subtracted from each score, of the deviations.