# Describing Statistics

## Reliability

In statistics, the word reliability is the overall consistency of a given measure. Under consistent conditions, if a measure provides results that are similar then it is said to have a reliability (Leech 26). In another definition, reliability may also be described as the characteristic of a given set of test scores relating to the number of random errors starting from the process of measurement that might get attached in the scores (Kim and Choong-Rak 323). Consistency, reproducibility, and accuracy are terms for highly reliable scores. Furthermore, the same results would be produced by the testing process if it were repeated with different test takers ( Kim and Choong-Rak 324). Validity In statistics, Validity is defined as the degree to which a measurement, conclusion or concept is well-founded and corresponds precisely with the real world (Leech 27). A test is considered to be valid if it measures what it is intended to measure. In some cases, the validity of a measurement is an equivalent of its accuracy (Kim and Choong-Rak 325). Validity is helpful in the determination of the types of tests to use, as well as ensuring that researchers apply research methods that are not only cost-effective, and ethical, but also methods that correctly measure the construct, or idea in question (Leech 28).

## Bell Curve

A bell curve is an alternative term for normal distribution curve or Gaussian distribution in statistics (Lesser 9). The “bell curve” name emerges from the fact that the curve appears bell-shaped as shown in the figure below.

The mean for a bell curve is usually at the center, and the curve has only one peak or mode (Lesser 10). Additionally, a bell curve has a predictable standard deviation and is always symmetric (Lesser 11).

## Mean

The statistical mean refers to the average, or mean used for deriving the central tendency of a given set of data (Weiers, Gray, and Lawrence 294). The determination of the mean involves the addition of all the data points in a given population and then dividing the sum by the number of points (Weiers, Gray, and Lawrence 296). The number that results from the division is the average or the mean (Weiers, Gray, and Lawrence 294). The statistical mean has a broad range of applications in different types of experimentation (Lesser 12). The use of mean in the calculation of various statistical measurements helps in eliminating random errors, as well as assist in deriving more accurate result that those derived from single experiments (Lesser 13). Additionally, the statistical mean is essential in interpreting statistical data since it includes all the items in the data set. However, the use of statistical mean has a disadvantage of being affected by the extreme values or points in the data set, which may intern make it biased (Weiers, Gray, and Lawrence 299).

## Standard Deviation

In statistics, standard deviation refers to the measure of deviation found by extracting the square root of the average (Mean) of the observed values’ squared deviation from their mean or average in a frequency distribution (Ober 2367). That is, the standard deviation of a given data set is the square root of its variance. It is a measure used in quantifying the amount of dispersion or variation of a set of data values (Ober 2368). If a standard deviation is low, then it shows that the data points appear to be close to the mean or the expected value of the set. A high standard deviation shows that the data points are distributed or spread out over a broad range of values (Weiers, Gray, and Lawrence 309).

## Standard Scores

The standard score, in statistics, refers to the standard deviations' signed number by which the value of data point or observation is above the mean (average) value of what is being measured or observed (Kim and Choong-Rak 25). The observed values that appear above the mean have positive standard scores, while those below the mean have standard scores that are negative (Weiers, Gray, and Lawrence 313). The standard score is a quantity that is dimensionless and obtained by dividing the difference between the population mean and the individual raw scores by the population standard deviation. Standard scores, also termed as z-scores or z-values are frequently used to compare a given observation a standard normal deviation (Weiers, Gray, and Lawrence 315).

## Scaled Scores

A scaled score refers to a conversion of the raw score of a student on a test, or a conversion of the test score to a common scale that enables or allows for numerical comparison between the students (Ober 2369). Since a test always has multiple versions, the use of the scale helps in controlling slight variations from one form or version of the test to another. The scale scores get reported in four different performance levels on the Standard Base Assessment, and levels include the beginning steps, nearing proficiency, proficient, and advanced level (Weiers, Gray, and Lawrence 325).

## T-Scores

T-scores, in statistics, refers to the ratio of how far an estimated parameter is from its standard error and its notional value (Weiers, Gray, and Lawrence 329). T-scores is usually used in hypothesis testing to show individuals how distant or far their score is from the known mean. T scores have a standard deviation of 10 and a mean of 50. Therefore, if the raw score of a student gets converted to a T-score, for example, and the T score becomes 70, then it means that the student's score is 20 points above the mean (Weiers, Gray, and Lawrence 330).

## Percentiles

In statistics, a percentile is a measure that indicates the value below which a given observations’ percentage fall in a group of observations (Leech 30). For example, the 30th percentile refers to the score or value below which 30 percent of the observations may be found (Leech 32). The use of percentiles is essential in reporting of scores from various norm-referenced tests. For instance, if a value or score is at the 56th percentile, where 56 refers to the percentile rank, then it is equal to the score below which 56 percent of the observations may be found (Weiers, Gray, and Lawrence 346).

## Works Cited

Kim, Jae-Wan and Choong-Rak Kim. "Basic Statistics In Quantile Regression." Korean Journal of Applied Statistics 25.2 (2012): 321-330. Web.

Leech, Nancy L. "Statistics Poker: Reinforcing Basic Statistical Concepts." Teaching Statistics 30.1 (2008): 26-28. Web.

Lesser, Lawrence M. "Simple Data Sets For Distinct Basic Summary Statistics." Teaching Statistics 33.1 (2011): 9-11. Web.

Ober, Pieter Bastiaan. "Basic Statistics." Journal of Applied Statistics 38.10 (2011): 2367-2369. Web.

Weiers, Ronald M, J. Brian Gray, and Lawrence H Peters. Introduction To Business Statistics. 1st ed. Australia: South-Western Cengage Learning, 2011. Print.

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