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## Introduction to Statistics

Statistics is the emerging domain which is increasingly used by the business firms to improve their business performance. In the current report, different methods that are used by the researchers in respect to computing confidence interval for median are discussed in detail. Along with this, formula of the methods like bootstrap normal, bootstrap percentile, binomial and Quantile optimality ratio is also given in the report. At end of the report, conclusion section is prepared and in this way entire research work is carried out.

### Bootstrap approach (Percentile)

Bootstrap method is the one of the popular method that is used in statistics by the business managers. Under this method sample values are changed and new sample is prepared number of times. Percentile bootstrap method believe in making some assumptions in respect to distributions that are taken in to account. On the basis of observation and estimation new samples are prepared in the bootstrap method. These estimated distribution of samples are used to calculate the confidence interval high and low values (Bland, 2015). Not only is this estimated distribution of sample also used to compute standard errors, bias and testing a hypothesis. The main principal for estimating sample distribution is resampling of values. Mentioned method was developed by Julian Simon. Percentile bootstrap is used to calculate compute the confidence interval for any quantity. Even sampling distribution can be estimated analytically or not percentile bootstrap method can be used by an individual (The percentile bootstrap, 2017). It is not necessary that every time through percentile bootstrap method right value of confidence interval will be computed. There are some methods that can give better result than percentile bootstrap method. Some of these alternative methods are percentile –t etc. It can be said that percentile interval of median is the one of the most powerful tool that is available to the research analysts. This is because in this method in terms of percentage range is computed which is assumed as lower and upper interval value at specific confidence interval which is 95% or any other percentage. It can be said that there is a great importance of the mentioned method for the analysts as it is helping them to calculate lower and upper value in interval in respect to median value.

### Bootstrap approach (Normal)

It is another method that is used on large scale by the business managers to compute lower and upper range of data. Normal approximation method calculate the approximate value of the standard error that is related to the analyzed data. This is done by using sampling distribution while is done by bootstrapping of the data. Bootstrapping is the approach by using which resampling of data is done by the researcher. In this approach upper and lower class interval for median is calculated by using Z distribution table (Hollander, Wolfe and Chicken, 2015). There is a slight difference between normal bootstrapping method and percentile method. In the latter sort of method frequency histogram is taken in to account. This frequency histogram is related to the m statistics which is calculated by using bootstrap sample. It can be said that there is a significant importance of the bootstrap method because in same normal data is taken in to account. It can be said that bootstrapping is the any method that heavily believe on the replacement of the sample. By doing so it is ensured that sample will be collected in proper way. It can be said that there is a huge significance of the bootstrap normal method for the business managers. Median class interval must be computed by using bootstrap because by using same mentioned one can be calculated in best way. There are number of ways in which confidence interval for median value can be computed by an individual and it depends on the individual that which method it find most important for calculation purpose. It can be said that there are is a significant importance of the bootstrap approach method for the individuals. ### Binomial

It is observed that calculation process for binomial distribution is very tough in case of median. This is because it is observed that it is very difficult task to algebraically manipulate the median relative to mean value. Binomial method is used to compute the confidence interval for median value. It is very simple to compute the confidence interval for median value. It is observed that intervals tend to be very conservative in case of binomial method. It is well known fact that median is the middle value that exist in the entire data set. This value divide entire data in to equal parts (Nielsen and et.al., 2014). There may be different trends in these parts. It is very important to identify these parts and analyze same because by doing so trends and patterns in which values of variables are moving can be identified. In order to find out values of confidence interval in the formulas that is given below it is necessary to identify the values of j and k. In the formula given above it can be observed that n refers to the sample size. Sample size may be any and selection of same depends on the researcher that is undertaking a research study. Q is indicating the proportion that one intends to find out a confidence interval. First of all in order to compute lowest class interval one needs to identify sample size and then same is multiplied with the proportion. Thereafter, from the z table one need to identify relevant value of confidence interval. Then one need to do square root of multiplied value that is done by multiplying number of sample units by proportion. Computed value is multiplied by 1- proportion percentage. This approach is followed for computing low value of confidence interval. After computing lower value of confidence interval upper value of same is computed. In this regard number of sample units is multiplied to proportion first of all and on same 1.96 is added which is the z value at 95% confidence interval. Thereafter, square root of multiplied value is taken like is done in case of lower value of confidence interval. Multiplied value the output of number of sample units and proportion. Then calculated value is multiplied by 1-q. In this way upper value of confidence interval is calculated in the binomial method.

### Quantile optimality ratio

It is the one of the most important method that is used to compute class interval in respect to median. Under this method confidence interval for Quantile is needed for proper estimate of Quantile density. Optimal bandwidth in respect to Kernel estimation of the Quantile density more or less depends on the quintile optimality ratio (Ferreri and et.al., 2014). It must be noted that Quantile optimality ratio is distribution free in nature. It must be noted that one family may do well in respect to other group that is having same shape. Data can be moddled by using a generalized lambda distribution. It can be observed that Quantile optimality ratio is the one of the most powerful tool that is used to analyze the data by the business firms. Formula that must be consider to compute lower and upper class interval is given below. It can be observed that there is a slight difference between the upper and lower class interval formulas. In case of lower class interval formula 1- alpha value is divided by 2. Whereas, in case of upper class interval 1- alpha value is not divided by any value. It can be said that there is a high degree of similarity between lower and upper class interval formula. Thus, it can be said that there is a great significance of the quartile optimality ratio (De Azambuja and et.al., 2014). There are number of the methods that can be used by the business firm for computing upper and lower interval of confidence interval of median. It depends on the individual that which method it think is best for data analysis.

#### Conclusion

On the basis of above discussion it is concluded that statistics is the vast domain and in it there are number of tools and methods that can be used to analyze the data. There are number of alternatives for doing a specific calculation. For computing class interval for median value there are number of methods that can be used by the analysts like bootstrap normal, bootstrap percentile, binomial and Quantile optimality ratio. All these approaches compute class interval in different manner and it depend on the researcher that which method it assumed more appropriate for research.

#### References

• Bland, M., 2015. An introduction to medical statistics. Oxford University Press (UK).
• De Azambuja, E. and et.al., 2014. Trastuzumab-associated cardiac events at 8 years of median follow-up in the Herceptin Adjuvant trial (BIG 1-01). Journal of clinical oncology.
• Ferreri, A. J. and et.al., 2014. MATILDE chemotherapy regimen for primary CNS lymphoma Results at a median follow-up of 12 years. Neurology.
• Hollander, M., A Wolfe, D. and Chicken, E., 2015. The One‐Sample Location Problem. Nonparametric Statistical Methods. Third Edition.
• Nielsen, J. B. and et.al., 2014. Risk prediction of cardiovascular death based on the QTc interval: evaluating age and gender differences in a large primary care population. European heart journal.

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