File Name: measures of central tendency and dispersion .zip
In statistics , a central tendency or measure of central tendency is a central or typical value for a probability distribution.
Measures of dispersion
Statistics Statistics deal with the collection, classification, description, presentation, and analysis interpretation of data numerical information. They are based upon observations or measurements of data. Statistics is inductive which implies specific observations and measurements yield a more general conclusion.
Statistics rely upon some notion of repetition. It follows, estimates can be derived, and variation and uncertainty of an estimate understood from repeated observations.
Statistics are often used to describe and summarize data. Probabilities or estimates of outcomes can also be determined for an event at a given location within specified limits. Statistics is sometimes referred to as a study of variation in data sets. Statistics allow us to make inferences conclusions most often based upon known and accepted facts from a sample, based upon a population of numerical data. Sampling statistics represent a portion of the total population set of data.
Populations are groups or aggregates of data. An estimate statistic is a property of a sample drawn at random by chance from a population. Estimates are expressed by roman letters. A sample standard deviation is symbolized by the letter s, mean is xbar, and variance is s2. A more detailed discussion on these measures will be addressed in later sections.
A data set may consist of, observations, variables, and variates. Inferential Statistics are based upon probability theory. Standard deviation is s, mean is xbar, and variance is s2. Estimates are values based upon confidence intervals. If the functional relationship is not known, causal conclusions cannot be inferred.
Values which belong to a continuous series include; height, weight, chronological time, discharge, velocity, etc. An eight bit spectral digital number range is from 0 to The size of a family 3 implies the exact size of the group.
Other examples include school enrollment and number of books in a library. Variables must be exhaustive and mutually exclusive. Interval data may transformed to ratio data by subtracting the differences of variates which eliminates or cancels out the arbitrary origin. A measurement of how close positions are clustered. Precision is based on a relative reference, e.
A subjective parameter, e. Reliability — How consistent, repeatable, or stable is the data over changes in spatial pattern over time? Basic Statistical Properties Constant — A property common to all members of a group. The total area under the curve is always infinite since the curve never intersects with the x axis, but for convenience the total area is taken as unity 1. The Normal Curve Figure 1 is written in standard score Z scores form with a mean equal to 0, variance equal to 1, and standard deviation equal to 1.
Figure 1. A normal distribution illustrating the area percentages for plus or minus three standard deviations. The area under the curve is taken as unity 1. On the left side of the curve, -1 standard deviation unit also accounts for 0. Therefore, plus or minus 1 standard deviation unit combined, accounts for approximately 0.
When this value is combined with plus or minus 1 standard deviation unit, 0. Three standard deviation units would only account for an additional 0. Standard Scores Z scores are derived as a transformation from raw scores variates to standard deviation units which are used to compare a score variates with a collection of scores variates derived from different procedures e.
Position is considered rather than the magnitude and measurement of units of scores. The discussions for measures of central tendency and measures of dispersion are presented later in this set of statistics notes. This will give a value of area between the mean and a Z score. If the raw score is greater than the mean, add the Z score to 50 to obtain the percentile rank. If the raw score is lower than the mean, subtract the Z score from A frequency distribution shows the number of times each value occurs, and arranges scores from lowest to highest.
Given a frequency distribution of values for example brightness values of 56, 57, 67, 99, , a histogram plots frequency counts on the vertical axis, and variables brightness values in this case are plotted on the horizontal axis. The Ogive or cumulative frequency Cf plot is a continuous count of frequencies for each BV at or below a given level. A typical scatterplot plotting duration and waiting time between eruptions of Old Faithful geyser, Yellowstone National Park, Wyoming.
A percentile rank is a percentile corresponding to a raw score in which as an example, if one is in the 90 percentile percentile rank , it would be interpreted as 90 percent of the scores are at or below this value, while 10 percent of the scores are above this value.
To obtain the percentile rank for a given score, e. Calculate the lower true limit of the score 67 by subtracting 0. Subtract the lower limit Multiply the result by the frequency of scores with a value of 67 Divide the result by the width of the class interval 1 in this case. Add the result to the cumulative frequency Divide the result by the total number of frequencies In a given distribution of brightness values frequency distribution , it is important to recognize a variety of properties about the distribution.
The four properties include: mode, median, arithmetic mean and deviation from the mean. It follows the mean is a measure of central location in the least square sense. Median — The point on the number scale such that half of the observations fall above it and half below it.
If the frequency of occurrences is equal for each value, there is no mode. Where two values have equal frequency, the mode is determined by adding the brightness values of the two that occur equally, and dividing by the total number of repetitive values 2.
The mode represents the highest point on a curve histogram. In a normal distribution symmetric bell curve, Figure 3 , the mean, mode, and median are the same values. Figure 3. A normal curve illustrating the relationship between the mean, median and mode. When a distribution is not normal, it results in the tail extending long to the right positively skewed toward the high end of the distribution.
When the tail trends to the left it is negatively skewed toward the low end of the distribution. In a positively skewed distribution the mean, median, and mode are distributed as illustrated below in Figure 4.
A positively skewed distribution illustrating the relationship between the mean, median, and mode. Figure 5. A negatively skewed distribution illustrating the relationship between the mean, median, and mode. What should become apparent is that a skewed distribution places the mean and median toward the end opposite the direction of the tail with the most frequently occurring scores mode.
In a negative distribution with treatments e. The apparent position of the mean and median in this negative distribution is shifted more towards the higher income treatments which gives one the impression a median value is regarded as being more controlled by income than frequency.
Recall in a normal distribution Figure 2 , the mean, median, and mode all fall in line with the same treatment value. A preferred method of dealing with negative signs would be to report a measure of variability as the standard deviation s which is a measure of variation in units of original measurements. Alternatively, a negative one If the standard deviation is 2. Correlation is a degree of relationship between variables. The closer an association between two variables approaches one, the higher the correlation.
No correlation between observations implies they are independent of each other; therefore, they are not correlated. In image processing, high correlations between two bands would suggest that using only one of the bands would account for a majority of the variability in the spectral values throughout the entire scene.
It follows correlation can be used to reduce the dimensionality of the data use less bands in a final classification to a more manageable number. This has important applications when using hyper-spectral data scenes with over spectral bands, since computer processing time could become very long the more bands that are used in the classification process.
Related Papers. Introduction to Statistics.
Average: It is a value which is typical or representative of a set of data. Averages are also called Measures of Central Tendency. Simple to calculate. It should be easy to understand. Rigidly defined. Based on all items of observation. Least affected by extreme values.
Measures of dispersion
While measures of central tendency are used to estimate "normal" values of a dataset, measures of dispersion are important for describing the spread of the data, or its variation around a central value. Two distinct samples may have the same mean or median, but completely different levels of variability, or vice versa. A proper description of a set of data should include both of these characteristics. There are various methods that can be used to measure the dispersion of a dataset, each with its own set of advantages and disadvantages. Standard Deviations Away From Mean.
The first exercise focuses on the research design which is your plan of action that explains how you will try to answer your research questions. Exercises two through four focus on sampling, measurement, and data collection.
A measure of central tendency is an important aspect of quantitative data. Three of the many ways to measure central tendency are the mean , median and mode. The sample mean is a statistic and a population mean is a parameter. Review the definitions of statistic and parameter in Lesson 0. Is this a problem?
Symbolically C. The distribution of the cost of production in rupees of a quaintal of wheat in 50 farms is as follows: 2. Yes, we agree with the statement. Save my name, email, and website in this browser for the next time I comment. Question Coefficient of variation is a percentage expression of standard deviation.
Она постучала пальцем по кипе документов: - Вот твоя жизнь, Чед Бринкерхофф.
Спустя несколько секунд Соши преобразовала на экране, казалось бы, произвольно набранные буквы. Теперь они выстроились в восемь рядов по восемь в каждом. Джабба посмотрел на экран и в отчаянии всплеснул руками. Новый порядок букв показался не более вразумительным, чем оригинал.
Чтобы скрыть свою маленькую тайну. Стратмор сохранял спокойствие. - И что же это за секрет.