You are asking about what statistical method to use on your test results conducted on 200 participants to get one descriptive value.  You really only have two choices - a mean or a median.  The mean will allow you to talk about the average from this group, and the median will allow you to talk about the point at which 50% of the group performed.

    Neither of these will get you what you seem to actually want, which is to be able to point to a value that is a "norm" for your assessment.  That is because the idea of a "norm" has to do with the group from whom you gathered the information. Taking the mean as one option:

- If I have assessment results from a nationally representative sample, I can talk about the mean score as being the average that reflects the performance of the nation being represented by that sample. 

- If I have assessment results from a sample of participants who have been randomly assigned to not receive a given treatment, I can talk about the mean score as being the average that reflects the performance of participants who have not had the treatment (the control group).

- If I have assessment results from a group of participants who are all in 10th grade in one school, I can talk about the mean score as being the average that reflects the performance of 10th graders in that school (and really, no one else).

      So, who are your 200 participants and what group do they represent (and by what generalizability argument can you make that case)?

Julia's answer was very good. I will just augment it a bit, and hope that it is helpful.

There is also the mode: a score achieved by the largest number of participants.

If you are interested in means, you are probably interested in the arithmetic mean. There are other choices that you can look at by visiting: http://en.wikipedia.org/wiki/Mean.

However, there are also other values that you can calculate that will help you understand your data. There is the variance which is a measure of how spread out individual observations are from the mean. The standard deviation (SD) is just the square root of the variance. 68.2% of all new observations will be within 1 standard deviation of the mean, and 95.6% will be within 2 standard deviations (assuming your data are normally distributed).

Skewness: Normally distributed data have a mean value with lots of observations close to the mean and fewer observations as one gets further from the mean. Also, Normally distributed data have an equal frequency of values greater than the mean versus less than the mean. In skewed data, this is not true. In Julia's 10th grade example, one might expect that most children are moderately athletic, but they also watch TV, play video games, etc... Yet there will be a few that are very active, and these will have much better endurance and motor skills than the rest. So the distribution will be skewed towards being less athletic. One calculation for skewness is little more than subtracting the median from the mean.

Kurtosis: Normally distributed data have a mean value with lots of observations close to the mean and fewer observations as one gets further from the mean. The description of how quickly I get fewer observations is kurtosis. Kurtosis has also been described as a measure of peakedness, or the fatness of the tails of the distribution.

Wikipedia has some very helpful figures showing standard deviation, skewness, and kurtosis. Just use these as key words in Wikipedia to get to relevant page.

To Julia's final question, I would ask: what is your goal/question/problem that has driven you to gather these data?

Any good textbook on medical statistics will show you how to create reference ranges for a variable.

Please refer the book below:

Statistics for Experimenters: Design, Innovation, and Discovery by George E. P. Box, J. Stuart Hunter, William G. Hunter 

You could consult the statistics department.

There is probably an introduction to statistics course taught, and you could use the textbook that they use.

Mostly, the favorite statistics book is a matter of which one you are forced to struggle through. At this level, I don't think that my favorite book is in print anymore. The best approach also depends on where or how far you want to go. If you are interested in ANOVA, regression, and multivariate analysis, then you will use all of an introductory text and more. If your interest in statistics starts and ends with this question, then even an introductory text will have a great deal of material that exceeds your level of interest. 

So, do you care that the value for the sprint test may be related to the score for the basketball throw test? Are there other data? One might hypothesize that age, and background may influence the results. All of these sorts of questions lead to more complex topics in data analysis.

Reference values are the values sampled from a reference population to calculate reference Limits or the refernce interval. So, I assume you have asked for reference Limits/refernce interval.

The simplest method to achieve reference limits are the 2.5 and 97.5 percentile in case of two-sided reference Limits (reference interval), and the 95% percentile (5% percentile) in case of a one sided reference limit.

Non parametric methods (like percentiles) are the best choice because prerequisites of normal distribution are often violated, especially at the tails of the Distribution, even after a transformation to achieve normal Distribution.

The disadvantage of the methodology are broader confidence intervals for estimates of refernce limits compared with parametric methods.

Literature: CLSI Guideline C28-A3 (Guideline was established for clinical lab)

Look for papers from Horn PS ("Horn PS, reference interval")

Best wishes

Thomas Keller

ACOMED Statistik, Leipzig, Germany

With the clarification provided by Thomas, you have 200 reference values for both the sprint test and the basketball throw test. The normative value is the ideal standard. That might be the mean of the reference population, or it might be the top 1% (or whatever percentage you choose). You probably cannot achieve either because you cannot sample the entire population, but you can approximate these values by calculating the sample mean or the top percentage of your sample.

The "reference population" idea is important. You always need to identify the population that you sampled. Any limits on that sample (racial, geographical, gender, socio-economic, etc...) also limit your population. Your intent in this case is largely irrelevant. In my research my intent is that my results apply to all insects of the same species that I study. However, the population that I sample is really those insects near my lab in Florida, and the probability that my results apply elsewhere declines with increasing distance from my lab. So all of us work with reference values, though we don't call them such. So "reference limits" are called confidence limits/intervals in my research and my statistics books.

Give Thomas's clarification, I reread the original question, and I am not sure what Paul is looking for. Could Paul please rephrase the question in light of the answers already given?

Dear all,

Thank you so much for the valuable suggestions. We are working among children with hemophilia. We wanted to check if the children can perform fitness test as competent to the normal children. So we have collected mean scores on the two test using convenient sampling. We wanted to construct a reference value using the normative score and test if our children are able to perform as normal. However, we it cant be so we wanted to have 4 values mostly based on the sd of the mean.

I came across GAMLSS - Generalized additive model for location, scale and shape and wanted to know is it a right model to device the normative value or to use percentile curves or using standard deviation to construct a normative score.

Do kindly suggest me. Thank you so much for the time you have taken to help in our survey.

The GAMLSS web site (GAMLSS.Org) gives the following definition: "GAMLSS are (semi) parametric univariate regression models, where all the parameters of the assumed distribution for the response can be modelled as additive functions of the explanatory variables." This sounds like GAMLSS is designed for use with a continuous dependent variable. Your dependent variable is categorical. So I suspect that this approach will not work.

Your basic question is relatively simple. That means that there are a large number of ways that you could go about answering the question depending on what assumptions you are willing to make.

You have a qualitative dependent variable (Health), and two quantitative independent variables (Sprint, and Ball).

1) Plot the data. If you plot the values of sprint on the y-axis, and health on the x-axis, do you get two distinct groups? Do the 95% confidence intervals for the means of sprint for each group of health overlap? Same for ball?

2) Given the plot of the data, is it reasonable to think that there is a difference between the hemophiliacs versus healthy children?Take any statistical package you can. Run a GLM (general linear model). Run three models: health=sprint, health=ball, health=sprint+ball+sprint*ball.

     a) are the ANOVA models significant? With only two treatments the model F-test is all you need. Multiple comparison procedures are unnecessary, since there is only one comparison.

    b) do they explain a large proportion of the observed variance (r-squared value is close to 1)?

    c) are the residuals well behaved?

    d) If the answer to either of the first two is no, then you might need to transform the data, or there really is no difference large enough to be detected by a sample of 200. Look at the residuals, they may help you figure out the problem.

3) I don't like discouraging the exploration of data and statistical methods. So you could try GAMLSS and see what you get. There are other choices: maximum likelihood, and randomization tests come to mind. Bayesian methods may also work. However, my guess is that all of this is overkill. An excessive use of complex statistical  designs where the simplest approach is more than sufficient.

Note: This is the answer that I have given your response. However I am not clear what you were trying to say in the last sentence of the first paragraph. I note that there are some terminology problems, that may be part of the problem: e.g. one cannot collect mean scores. One cannot construct a reference value, at least not by the definitions that Thomas provided. One collects observations, data, or reference values. Means are calculated from these data.