Pearson correlation coefficient
In statistics, the Pearson correlation coefficient referred to as Pearson's r, the Pearson product-moment correlation coefficient or the bivariate correlation, is a measure of the linear correlation between two variables X and Y. According to the Cauchy–Schwarz inequality it has a value between +1 and −1, where 1 is total positive linear correlation, 0 is no linear correlation, −1 is total negative linear correlation, it is used in the sciences. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s and for which the mathematical formula was derived and published by Auguste Bravais in 1844.. The naming of the coefficient is thus an example of Stigler's Law. Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations; the form of the definition involves a "product moment", that is, the mean of the product of the mean-adjusted random variables. Pearson's correlation coefficient when applied to a population is represented by the Greek letter ρ and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient.
Given a pair of random variables, the formula for ρ is: where: cov is the covariance σ X is the standard deviation of X σ Y is the standard deviation of Y The formula for ρ can be expressed in terms of mean and expectation. Since cov = E , the formula for ρ can be written as where: σ Y and σ X are defined as above μ X is the mean of X μ Y is the mean of Y E is the expectation; the formula for ρ can be expressed in terms of uncentered moments. Since μ X = E μ Y = E σ X 2 = E = E − 2 σ Y 2 = E = E − 2 E = E = E − E E , the formula for ρ can be written as ρ X, Y = E − E E E − 2 E −
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be undefined. For a unimodal distribution, negative skew indicates that the tail is on the left side of the distribution, positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value means. Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side; these tapering sides are called tails, they provide a visual means to determine which of the two kinds of skewness a distribution has: negative skew: The left tail is longer. The distribution is said to be left-skewed, left-tailed, or skewed to the left, despite the fact that the curve itself appears to be skewed or leaning to the right. A left-skewed distribution appears as a right-leaning curve.
Positive skew: The right tail is longer. The distribution is said to be right-skewed, right-tailed, or skewed to the right, despite the fact that the curve itself appears to be skewed or leaning to the left. A right-skewed distribution appears as a left-leaning curve. Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the numeric sequence, whose values are evenly distributed around a central value of 50. We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, e.g.. We can make the sequence positively skewed by adding a value far above the mean, e.g.. The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, for positive skew. In the older notion of nonparametric skew, defined as / σ, where μ is the mean, ν is the median, σ is the standard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than the median, while negative/left nonparametric skew means the mean is less than the median.
However, the modern definition of skewness and the traditional nonparametric definition do not in general have the same sign: while they agree for some families of distributions, they differ in general, conflating them is misleading. If the distribution is symmetric the mean is equal to the median, the distribution has zero skewness. If the distribution is both symmetric and unimodal the mean = median = mode; this is the case of a coin toss or the series 1,2,3,4... Note, that the converse is not true in general, i.e. zero skewness does not imply that the mean is equal to the median. A 2005 journal article points out: Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew, left of the median under left skew; this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal.
Such distributions not only contradict the textbook relationship between mean and skew, they contradict the textbook interpretation of the median. The skewness of a random variable X is the third standardized moment γ1, defined as: γ 1 = E = μ 3 σ 3 = E 3 / 2 = κ 3 κ 2 3 / 2 where μ is the mean, σ is the standard deviation, E is the expectation operator, μ3 is the third central moment, κt are the tth
A population pyramid called an "age-sex pyramid", is a graphical illustration that shows the distribution of various age groups in a population, which forms the shape of a pyramid when the population is growing. Males are conventionally shown on the left and females on the right, they may be measured by raw number or as a percentage of the total population; this tool can be used to age of a particular population. It is used in ecology to determine the overall age distribution of a population. Population pyramids contain continuous stacked-histogram bars, making it a horizontal bar diagram; the population size is depicted on the x-axis. The size of the population can either be measured as a percentage of the total population or by raw number. Males are conventionally shown on females on the right. Population pyramids are viewed as the most effective way to graphically depict the age and distribution of a population because of the clear image these pyramids represent. A great deal of information about the population broken down by age and sex can be read from a population pyramid, this can shed light on the extent of development and other aspects of the population.
The measures of central tendency, mean and mode, should be considered when assessing a population pyramid. Since the data is not accurate. For example, the average age could be used to determine the type of population in a particular region. A population with an average age of 15 would have a young population compared to a population that has an average age of 55, which would be considered an older population, it is important to consider these measures because the collected data is not accurate. The mid-year population is used in calculations to account for the number of births and deaths that occur. A population pyramid gives a clear picture of how a country transitions from high fertility to low fertility rate; the broad base of the pyramid means the majority of population lies between ages 0–14, which tells us that the fertility rate of the country is high and above population sub-replacement fertility level. The older population is declining over time due to a shorter life expectancy of sixty years.
However, there are still more females than males in these ranges since women have a longer life expectancy. As reported by the Proceedings of the National Academy of Sciences, women tend to live longer than men because women do not partake in risky behaviors. Weeks' Population: an Introduction to Concepts and Issues, considered that the sex ratio gap for the older ages will shrink due to women's health declining due to the effects of smoking, as suggested by the United Nations and U. S. Census Bureau. Moreover, it can reveal the age-dependency ratio of a population. Populations with a big base, young population, or a big top, an older population, shows that there is a higher dependency ratio; the dependency ratio refers to. According to Weeks' Population: an Introduction to Concepts and Issues, population pyramids can be used to predict the future, known as a population forecast. Population momentum, when a population's birth rates continue to increase after replacement level has been reached, can be predicted if a population has a low mortality rate since the population will continue to grow.
This brings up the term doubling time, used to predict when the population will double in size. Lastly, a population pyramid can give insight on the economic status of a country from the age stratification since the distribution of supplies are not evenly distributed through a population. In the demographic transition model, the size and shape of population pyramids vary. In stage one of the demographic transition model, the pyramids have the most defined shape, they have the ideal big skinny top. In stage two, the pyramid starts to widen in the middle age groups. In stage three, the pyramids start to look similar in shape to a tombstone. In stage four, there is a decrease in the younger age groups; this causes the base of the widened pyramid to narrow. Lastly, in stage five, the pyramid starts to take on the shape of a kite as the base continues to decrease; the shape of the population is dependent upon. More developed countries can be found in stages three four and five while the least developed countries have a population represented by the pyramids in stages one and two.
Each country will have unique population pyramids. However, population pyramids will be defined as the following: stationary, expansive, or constrictive; these types have been identified by the mortality rates of a country. "Stationary" pyramid A pyramid can be described as stationary if the percentages of population remains constant over time. Stationary population is when a population contains equal birth rates and death rates."Expansive" pyramid A population pyramid, wide at the younger ages, characteristic of countries with high birth rate and low life expectancy. The population is said to be fast-growing, the size of each birth cohort gets larger than the size of the previous year."Constrictive" pyramid A population pyramid, narrowed at the bottom. The population is older on average, as the country has long life expectancy, a low death rate, but a low birth rate. However, the percentage of younger population are low, this can cause issues with dependency ratio of
A pie chart is a circular statistical graphic, divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice, is proportional to the quantity. While it is named for its resemblance to a pie, sliced, there are variations on the way it can be presented; the earliest known pie chart is credited to William Playfair's Statistical Breviary of 1801. Pie charts are widely used in the business world and the mass media. However, they have been criticized, many experts recommend avoiding them, pointing out that research has shown it is difficult to compare different sections of a given pie chart, or to compare data across different pie charts. Pie charts can be replaced in most cases by other plots such as the bar chart, box plot or dot plots; the earliest known pie chart is credited to William Playfair's Statistical Breviary of 1801, in which two such graphs are used. Playfair presented an illustration. One of those charts depicting the proportions of the Turkish Empire located in Asia and Africa before 1789.
This invention was not used at first. Minard's map, 1858 used pie charts to represent the cattle sent from all around France for consumption in Paris. Playfair thought, it has been said that Florence Nightingale invented it, though in fact she just popularised it and she was assumed to have created it due to the obscurity of Playfair's creation. A 3d pie chart, or perspective pie chart, is used to give the chart a 3D look. Used for aesthetic reasons, the third dimension does not improve the reading of the data; the use of superfluous dimensions not used to display the data of interest is discouraged for charts in general, not only for pie charts. A doughnut chart is a variant of the pie chart, with a blank center allowing for additional information about the data as a whole to be included. Doughnut charts are similar to pie charts; this type of circular graph can support multiple statistics at once and it provides a better data intensity ratio to standard pie charts. It does not have to contain information in the center.
A chart with one or more sectors separated from the rest of the disk is known as an exploded pie chart. This effect is used to either highlight a sector, or to highlight smaller segments of the chart with small proportions; the polar area diagram is similar to a usual pie chart, except sectors have equal angles and differ rather in how far each sector extends from the center of the circle. The polar area diagram is used to plot cyclic phenomena. For example, if the counts of deaths in each month for a year are to be plotted there will be 12 sectors all with the same angle of 30 degrees each; the radius of each sector would be proportional to the square root of the death count for the month, so the area of a sector represents the number of deaths in a month. If the death count in each month is subdivided by cause of death, it is possible to make multiple comparisons on one diagram, as is seen in the polar area diagram famously developed by Florence Nightingale; the first known use of polar area diagrams was by André-Michel Guerry, which he called courbes circulaires, in an 1829 paper showing seasonal and daily variation in wind direction over the year and births and deaths by hour of the day.
Léon Lalanne used a polar diagram to show the frequency of wind directions around compass points in 1843. The wind rose. Nightingale published her rose diagram in 1858. Although the name "coxcomb" has come to be associated with this type of diagram, Nightingale used the term to refer to the publication in which this diagram first appeared--an attention-getting book of charts and tables--rather than to this specific type of diagram. A ring chart known as a sunburst chart or a multilevel pie chart, is used to visualize hierarchical data, depicted by concentric circles; the circle in the center represents the root node, with the hierarchy moving outward from the center. A segment of the inner circle bears a hierarchical relationship to those segments of the outer circle which lie within the angular sweep of the parent segment. A variant of the polar area chart is the spie chart designed by Dror Feitelson; this superimposes a normal pie chart with a modified polar area chart to permit the comparison of two sets of related data.
The base pie chart represents the first data set with different slice sizes. The second set is represented by the superimposed polar area chart, using the same angles as the base, adjusting the radii to fit the data. For example, the base pie chart could show the distribution of age and gender groups in a population, the overlay their representation among road casualties. Age and gender groups that are susceptible to being involved in accidents stand out as slices that extend beyond the original pie chart. Square charts called Waffle Charts, are a form of pie charts that use squares instead of circles to represent percentages. Similar to basic circular pie charts, square pie charts take each percentage out of a total 100%, they are 10x10 grids, where each cell represents 1%. Despite the name, circles and other shapes may be used instead o
United States Census Bureau
The United States Census Bureau is a principal agency of the U. S. Federal Statistical System, responsible for producing data about the American people and economy; the Census Bureau is part of the U. S. Department of Commerce and its director is appointed by the President of the United States; the Census Bureau's primary mission is conducting the U. S. Census every ten years, which allocates the seats of the U. S. House of Representatives to the states based on their population; the Bureau's various censuses and surveys help allocate over $400 billion in federal funds every year and it helps states, local communities, businesses make informed decisions. The information provided by the census informs decisions on where to build and maintain schools, transportation infrastructure, police and fire departments. In addition to the decennial census, the Census Bureau continually conducts dozens of other censuses and surveys, including the American Community Survey, the U. S. Economic Census, the Current Population Survey.
Furthermore and foreign trade indicators released by the federal government contain data produced by the Census Bureau. Article One of the United States Constitution directs the population be enumerated at least once every ten years and the resulting counts used to set the number of members from each state in the House of Representatives and, by extension, in the Electoral College; the Census Bureau now conducts a full population count every 10 years in years ending with a zero and uses the term "decennial" to describe the operation. Between censuses, the Census Bureau makes population projections. In addition, Census data directly affects how more than $400 billion per year in federal and state funding is allocated to communities for neighborhood improvements, public health, education and more; the Census Bureau is mandated with fulfilling these obligations: the collecting of statistics about the nation, its people, economy. The Census Bureau's legal authority is codified in Title 13 of the United States Code.
The Census Bureau conducts surveys on behalf of various federal government and local government agencies on topics such as employment, health, consumer expenditures, housing. Within the bureau, these are known as "demographic surveys" and are conducted perpetually between and during decennial population counts; the Census Bureau conducts economic surveys of manufacturing, retail and other establishments and of domestic governments. Between 1790 and 1840, the census was taken by marshals of the judicial districts; the Census Act of 1840 established a central office. Several acts followed that revised and authorized new censuses at the 10-year intervals. In 1902, the temporary Census Office was moved under the Department of Interior, in 1903 it was renamed the Census Bureau under the new Department of Commerce and Labor; the department was intended to consolidate overlapping statistical agencies, but Census Bureau officials were hindered by their subordinate role in the department. An act in 1920 changed the date and authorized manufacturing censuses every two years and agriculture censuses every 10 years.
In 1929, a bill was passed mandating the House of Representatives be reapportioned based on the results of the 1930 Census. In 1954, various acts were codified into Title 13 of the US Code. By law, the Census Bureau must count everyone and submit state population totals to the U. S. President by December 31 of any year ending in a zero. States within the Union receive the results in the spring of the following year; the United States Census Bureau defines four statistical regions, with nine divisions. The Census Bureau regions are "widely used...for data collection and analysis". The Census Bureau definition is pervasive. Regional divisions used by the United States Census Bureau: Region 1: Northeast Division 1: New England Division 2: Mid-Atlantic Region 2: Midwest Division 3: East North Central Division 4: West North Central Region 3: South Division 5: South Atlantic Division 6: East South Central Division 7: West South Central Region 4: West Division 8: Mountain Division 9: Pacific Many federal, state and tribal governments use census data to: Decide the location of new housing and public facilities, Examine the demographic characteristics of communities and the US, Plan transportation systems and roadways, Determine quotas and creation of police and fire precincts, Create localized areas for elections, utilities, etc.
Gathers population information every 10 years The United States Census Bureau is committed to confidentiality, guarantees non-disclosure of any addresses or personal information related to individuals or establishments. Title 13 of the U. S. Code establishes penalties for the disclosure of this information. All Census employees must sign an affidavit of non-disclosure prior to employment; the Bureau cannot share responses, addresses or personal information with anyone including United States or foreign government
A scatter plot is a type of plot or mathematical diagram using Cartesian coordinates to display values for two variables for a set of data. If the points are color-coded, one additional variable can be displayed; the data are displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis. A scatter plot can be used either when one continuous variable, under the control of the experimenter and the other depends on it or when both continuous variables are independent. If a parameter exists, systematically incremented and/or decremented by the other, it is called the control parameter or independent variable and is customarily plotted along the horizontal axis; the measured or dependent variable is customarily plotted along the vertical axis. If no dependent variable exists, either type of variable can be plotted on either axis and a scatter plot will illustrate only the degree of correlation between two variables.
A scatter plot can suggest various kinds of correlations between variables with a certain confidence interval. For example and height, weight would be on y axis and height would be on the x axis. Correlations may be negative, or null. If the pattern of dots slopes from lower left to upper right, it indicates a positive correlation between the variables being studied. If the pattern of dots slopes from upper left to lower right, it indicates a negative correlation. A line of best fit can be drawn in order to study the relationship between the variables. An equation for the correlation between the variables can be determined by established best-fit procedures. For a linear correlation, the best-fit procedure is known as linear regression and is guaranteed to generate a correct solution in a finite time. No universal best-fit procedure is guaranteed to generate a correct solution for arbitrary relationships. A scatter plot is very useful when we wish to see how two comparable data sets agree to show nonlinear relationships between variables.
The ability to do this can be enhanced by adding a smooth line such as LOESS. Furthermore, if the data are represented by a mixture model of simple relationships, these relationships will be visually evident as superimposed patterns; the scatter diagram is one of the seven basic tools of quality control. Scatter charts can be built in the form of marker, or/and line charts. For example, to display a link between a person's lung capacity, how long that person could hold his/her breath, a researcher would choose a group of people to study measure each one's lung capacity and how long that person could hold his/her breath; the researcher would plot the data in a scatter plot, assigning "lung capacity" to the horizontal axis, "time holding breath" to the vertical axis. A person with a lung capacity of 400 cl who held his/her breath for 21.7 seconds would be represented by a single dot on the scatter plot at the point in the Cartesian coordinates. The scatter plot of all the people in the study would enable the researcher to obtain a visual comparison of the two variables in the data set, will help to determine what kind of relationship there might be between the two variables.
For a set of data variables X1, X2... Xk, the scatter plot matrix shows all the pairwise scatter plots of the variables on a single view with multiple scatterplots in a matrix format. For k variables, the scatterplot matrix will contain k rows and k columns. A plot located on the intersection of i-th row and j-th column is a plot of variables Xj; this means that each row and column is one dimension, each cell plots a scatterplot of two dimensions. A generalized scatterplot matrix offers a range of displays of paired combinations of categorical and quantitative variables. A mosaic plot, fluctuation diagram, or faceted bar chart may be used to display two categorical variables. Other plots are used for one quantitative variables. Rug plot What is a scatterplot? Correlation scatter-plot matrix for ordered-categorical data – Explanation and R code Density scatterplot for large datasets
Central limit theorem
In probability theory, the central limit theorem establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution if the original variables themselves are not distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. For example, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the distribution of the average will be approximated by a normal distribution. A simple example of this is that if one flips a coin many times the probability of getting a given number of heads in a series of flips will approach a normal curve, with mean equal to half the total number of flips in each series.
The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions; the earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is now known as the de Moivre–Laplace theorem. In more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory, they all express the fact that a sum of many independent and identically distributed random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. Variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d.
Random variables with power law tail distributions decreasing as |x|−α − 1 where 0 < α < 2 will tend to an alpha-stable distribution with stability parameter of α as the number of variables grows. Let be a random sample of size n—that is, a sequence of independent and identically distributed random variables drawn from a distribution of expected value given by µ and finite variance given by σ2. Suppose we are interested in the sample average S n:= X 1 + ⋯ + X n n of these random variables. By the law of large numbers, the sample averages converge in probability and surely to the expected value µ as n → ∞; the classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number µ during this convergence. More it states that as n gets larger, the distribution of the difference between the sample average Sn and its limit µ, when multiplied by the factor √n, approximates the normal distribution with mean 0 and variance σ2.
For large enough n, the distribution of Sn is close to the normal distribution with mean µ and variance σ2/n. The usefulness of the theorem is that the distribution of √n approaches normality regardless of the shape of the distribution of the individual Xi. Formally, the theorem can be stated as follows: Lindeberg–Lévy CLT. Suppose is a sequence of i.i.d. Random variables with E = µ and Var = σ2 < ∞. As n approaches infinity, the random variables √n converge in distribution to a normal N: n → d N. In the case σ > 0, convergence in distribution means that the cumulative distribution functions of √n converge pointwise to the cdf of the N distribution: for every real number z, lim n → ∞ Pr = Φ, where Φ is the standard normal cdf evaluated at x. Note that the convergence is uniform in z in the sense that lim n → ∞ sup z ∈ R | Pr − Φ | = 0, where sup denotes the least upper bound of the se