In economics, the Gini coefficient (/ˈdʒiːni/ JEE-nee), sometimes called Gini index, or Gini ratio, is a measure of statistical dispersion intended to represent the income or wealth distribution of a nation's residents, and is the most commonly used measurement of inequality. It was developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper Variability and Mutability (Italian: Variabilità e mutabilità).
The Gini coefficient measures the inequality among values of a frequency distribution (for example, levels of income). A Gini coefficient of zero expresses perfect equality, where all values are the same (for example, where everyone has the same income). A Gini coefficient of 1 (or 100%) expresses maximal inequality among values (e.g., for a large number of people, where only one person has all the income or consumption, and all others have none, the Gini coefficient will be very nearly one). However, a value greater than one may occur if some persons represent negative contribution to the total (for example, having negative income or wealth). For larger groups, values close to one are very unlikely in practice. Given the normalization of both the cumulative population and the cumulative share of income used to calculate the Gini coefficient, the measure is not overly sensitive to the specifics of the income distribution, but rather only on how incomes vary relative to the other members of a population. The exception to this is in the redistribution of income resulting in a minimum income for all people. When the population is sorted, if their income distribution were to approximate a well-known function, then some representative values could be calculated.
The Gini coefficient was proposed by Gini as a measure of inequality of income or wealth. For OECD countries, in the late 20th century, considering the effect of taxes and transfer payments, the income Gini coefficient ranged between 0.24 and 0.49, with Slovenia being the lowest and Chile the highest. African countries had the highest pre-tax Gini coefficients in 2008–2009, with South Africa the world's highest, variously estimated to be 0.63 to 0.7, although this figure drops to 0.52 after social assistance is taken into account, and drops again to 0.47 after taxation. The global income Gini coefficient in 2005 has been estimated to be between 0.61 and 0.68 by various sources.
There are some issues in interpreting a Gini coefficient. The same value may result from many different distribution curves. The demographic structure should be taken into account. Countries with an aging population, or with a baby boom, experience an increasing pre-tax Gini coefficient even if real income distribution for working adults remains constant. Scholars have devised over a dozen variants of the Gini coefficient.
The Gini coefficient is a single number aimed at measuring the degree of inequality in a distribution. It is most often used in economics to measure how far a country's wealth or income distribution deviates from a totally equal distribution.
The Gini coefficient is usually defined mathematically based on the Lorenz curve, which plots the proportion of the total income of the population (y axis) that is cumulatively earned by the bottom x of the population (see diagram). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (marked A in the diagram) over the total area under the line of equality (marked A and B in the diagram); i.e., G = A/(A + B). It is also equal to 2A and to 1 − 2B due to the fact that A + B = 0.5 (since the axes scale from 0 to 1).
If all people have non-negative income (or wealth, as the case may be), the Gini coefficient can theoretically range from 0 (complete equality) to 1 (complete inequality); it is sometimes expressed as a percentage ranging between 0 and 100. In practice, both extreme values are not quite reached. If negative values are possible (such as the negative wealth of people with debts), then the Gini coefficient could theoretically be more than 1. Normally the mean (or total) is assumed positive, which rules out a Gini coefficient less than zero.
When the income (or wealth) distribution is given as a continuous probability distribution function p(x), the Gini coefficient is again half of the relative mean absolute difference:
Example: two levels of income
The most equal society will be one in which every person receives the same income (G = 0); the most unequal society will be one in which a single person receives 100% of the total income and the remaining N − 1 people receive none (G = 1 − 1/N).
While the income distribution of any particular country need not follow simple functions, these functions give a qualitative understanding of the income distribution in a nation given the Gini coefficient.
An informative simplified case just distinguishes two levels of income, low and high. If the high income group is a proportion u of the population and earns a proportion f of all income, then the Gini coefficient is f − u. An actual more graded distribution with these same values u and f will always have a higher Gini coefficient than f − u.
The proverbial case where the richest 20% have 80% of all income (see Pareto principle) would lead to an income Gini coefficient of at least 60%.
In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example, (taking y to mean the income or wealth of a person or household):
For a population uniform on the values y**i, i = 1 to n, indexed in non-decreasing order (y**i ≤ y**i+1):
Since the Gini coefficient is half the relative mean absolute difference, it can also be calculated using formulas for the relative mean absolute difference. For a random sample S consisting of values y**i, i = 1 to n, that are indexed in non-decreasing order (y**i ≤ y**i+1), the statistic:
is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like G, G(S) has a simpler form:
There does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient, like the relative mean absolute difference.
Discrete probability distribution
For a discrete probability distribution with probability mass function f ( yi ), i = 1 to n, where f ( yi ) is the fraction of the population with income or wealth yi>0, the Gini coefficient is:
- If the points with nonzero probabilities are indexed in increasing order (yi< yi+1) then:
- andThese formulae are also applicable in the limit as
Continuous probability distribution
When the population is large, the income distribution may be represented by a continuous probability density function f(x) where f(x) dx is the fraction of the population with wealth or income in the interval dx about x. If F(x) is the cumulative distribution function for f(x), then the Lorenz curve L(F) may then be represented as a function parametric in L(x) and F(x) and the value of B can be found by integration:
The Gini coefficient can also be calculated directly from the cumulative distribution function of the distribution F(y). Defining μ as the mean of the distribution, and specifying that F(y) is zero for all negative values, the Gini coefficient is given by:
The latter result comes from integration by parts. (Note that this formula can be applied when there are negative values if the integration is taken from minus infinity to plus infinity.)
The Gini coefficient may be expressed in terms of the quantile function Q(F) (inverse of the cumulative distribution function: Q(F(x)) = x)
Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated by using various techniques for interpolating the missing values of the Lorenz curve. If (X**k, Y**k) are the known points on the Lorenz curve, with the X**k indexed in increasing order (X**k – 1 < X**k), so that:
X**k is the cumulated proportion of the population variable, for k = 0,...,n, with X0 = 0, X**n = 1.
Y**k is the cumulated proportion of the income variable, for k = 0,...,n, with Y0 = 0, Y**n = 1.
Y**k should be indexed in non-decreasing order (Y**k > Y**k – 1)
If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and:
is the resulting approximation for G. More accurate results can be obtained using other methods to approximate the area B, such as approximating the Lorenz curve with a quadratic function across pairs of intervals, or building an appropriately smooth approximation to the underlying distribution function that matches the known data. If the population mean and boundary values for each interval are also known, these can also often be used to improve the accuracy of the approximation.
The Gini coefficient calculated from a sample is a statistic and its standard error, or confidence intervals for the population Gini coefficient, should be reported. These can be calculated using bootstrap techniques but those proposed have been mathematically complicated and computationally onerous even in an era of fast computers. Ogwang (2000) made the process more efficient by setting up a "trick regression model" in which respective income variables in the sample are ranked with the lowest income being allocated rank 1. The model then expresses the rank (dependent variable) as the sum of a constant A and a normal error term whose variance is inversely proportional to y**k;
Ogwang showed that G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculation of the jackknife estimate for the standard error. Giles (2004) argued that the standard error of the estimate of A can be used to derive that of the estimate of G directly without using a jackknife at all. This method only requires the use of ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improving with increasing sample size.
However it has since been argued that this is dependent on the model's assumptions about the error distributions (Ogwang 2004) and the independence of error terms (Reza & Gastwirth 2006) and that these assumptions are often not valid for real data sets. It may therefore be better to stick with jackknife methods such as those proposed by Yitzhaki (1991) and Karagiannis and Kovacevic (2000). The debate continues.
Generalized inequality indices
where p**j weights the units by their population share, and f(r**j) is a function of the deviation of each unit's r**j from 1, the point of equality. The insight of this generalised inequality index is that inequality indices differ because they employ different functions of the distance of the inequality ratios (the r**j) from 1.**
Gini coefficients of income distributions
Gini coefficients of income are calculated on market income as well as disposable income basis. The Gini coefficient on market income—sometimes referred to as a pre-tax Gini coefficient—is calculated on income before taxes and transfers, and it measures inequality in income without considering the effect of taxes and social spending already in place in a country. The Gini coefficient on disposable income—sometimes referred to as after-tax Gini coefficient—is calculated on income after taxes and transfers, and it measures inequality in income after considering the effect of taxes and social spending already in place in a country.
The difference in Gini indices between OECD countries, on after-taxes and transfers basis, is significantly narrower. For OECD countries, over 2008–2009 period, Gini coefficient on pre-taxes and transfers basis for total population ranged between 0.34 and 0.53, with South Korea the lowest and Italy the highest. Gini coefficient on after-taxes and transfers basis for total population ranged between 0.25 and 0.48, with Denmark the lowest and Mexico the highest. For United States, the country with the largest population in OECD countries, the pre-tax Gini index was 0.49, and after-tax Gini index was 0.38, in 2008–2009. The OECD averages for total population in OECD countries was 0.46 for pre-tax income Gini index and 0.31 for after-tax income Gini Index. Taxes and social spending that were in place in 2008–2009 period in OECD countries significantly lowered effective income inequality, and in general, "European countries—especially Nordic and Continental welfare states—achieve lower levels of income inequality than other countries."
Using the Gini can help quantify differences in welfare and compensation policies and philosophies. However it should be borne in mind that the Gini coefficient can be misleading when used to make political comparisons between large and small countries or those with different immigration policies (see limitations of Gini coefficient section).
Regional income Gini indices
According to UNICEF, Latin America and the Caribbean region had the highest net income Gini index in the world at 48.3, on unweighted average basis in 2008. The remaining regional averages were: sub-Saharan Africa (44.2), Asia (40.4), Middle East and North Africa (39.2), Eastern Europe and Central Asia (35.4), and High-income Countries (30.9). Using the same method, the United States is claimed to have a Gini index of 36, while South Africa had the highest income Gini index score of 67.8.
World income Gini index since 1800s
Taking income distribution of all human beings, worldwide income inequality has been constantly increasing since the early 19th century. There was a steady increase in the global income inequality Gini score from 1820 to 2002, with a significant increase between 1980 and 2002. This trend appears to have peaked and begun a reversal with rapid economic growth in emerging economies, particularly in the large populations of BRIC countries.
The table below presents the estimated world income Gini coefficients over the last 200 years, as calculated by Milanovic.
More detailed data from similar sources plots a continuous decline since 1988. This is attributed to globalization increasing incomes for billions of poor people, mostly in India and China. Developing countries like Brazil have also improved basic services like health care, education, and sanitation; others like Chile and Mexico have enacted more progressive tax policies.
Gini coefficients of social development
Gini coefficient is widely used in fields as diverse as sociology, economics, health science, ecology, engineering and agriculture. For example, in social sciences and economics, in addition to income Gini coefficients, scholars have published education Gini coefficients and opportunity Gini coefficients.
Gini coefficient of education
Education Gini index estimates the inequality in education for a given population. It is used to discern trends in social development through educational attainment over time. From a study of 85 countries by three Economists of World Bank Vinod Thomas, Yan Wang, Xibo Fan, estimate Mali had the highest education Gini index of 0.92 in 1990 (implying very high inequality in education attainment across the population), while the United States had the lowest education inequality Gini index of 0.14. Between 1960 and 1990, China, India and South Korea had the fastest drop in education inequality Gini Index. They also claim education Gini index for the United States slightly increased over the 1980–1990 period.
Gini coefficient of opportunity
Similar in concept to income Gini coefficient, opportunity Gini coefficient measures inequality of opportunity. The concept builds on Amartya Sen's suggestion that inequality coefficients of social development should be premised on the process of enlarging people's choices and enhancing their capabilities, rather than on the process of reducing income inequality. Kovacevic in a review of opportunity Gini coefficient explains that the coefficient estimates how well a society enables its citizens to achieve success in life where the success is based on a person's choices, efforts and talents, not his background defined by a set of predetermined circumstances at birth, such as, gender, race, place of birth, parent's income and circumstances beyond the control of that individual.
Gini coefficients and income mobility
In 1978, Anthony Shorrocks introduced a measure based on income Gini coefficients to estimate income mobility. This measure, generalized by Maasoumi and Zandvakili, is now generally referred to as Shorrocks index, sometimes as Shorrocks mobility index or Shorrocks rigidity index. It attempts to estimate whether the income inequality Gini coefficient is permanent or temporary, and to what extent a country or region enables economic mobility to its people so that they can move from one (e.g., bottom 20%) income quantile to another (e.g., middle 20%) over time. In other words, Shorrocks index compares inequality of short-term earnings such as annual income of households, to inequality of long-term earnings such as 5-year or 10-year total income for same households.
Shorrocks index is calculated in number of different ways, a common approach being from the ratio of income Gini coefficients between short-term and long-term for the same region or country.
A 2010 study using social security income data for the United States since 1937 and Gini-based Shorrocks indices concludes that income mobility in the United States has had a complicated history, primarily due to mass influx of women into the American labor force after World War II. Income inequality and income mobility trends have been different for men and women workers between 1937 and the 2000s. When men and women are considered together, the Gini coefficient-based Shorrocks index trends imply long-term income inequality has been substantially reduced among all workers, in recent decades for the United States. Other scholars, using just 1990s data or other short periods have come to different conclusions. For example, Sastre and Ayala, conclude from their study of income Gini coefficient data between 1993 and 1998 for six developed economies, that France had the least income mobility, Italy the highest, and the United States and Germany intermediate levels of income mobility over those 5 years.
Features of Gini coefficient
The Gini coefficient has features that make it useful as a measure of dispersion in a population, and inequalities in particular. It is a ratio analysis method making it easier to interpret. It also avoids references to a statistical average or position unrepresentative of most of the population, such as per capita income or gross domestic product. For a given time interval, Gini coefficient can therefore be used to compare diverse countries and different regions or groups within a country; for example states, counties, urban versus rural areas, gender and ethnic groups. Gini coefficients can be used to compare income distribution over time, thus it is possible to see if inequality is increasing or decreasing independent of absolute incomes.
Anonymity: it does not matter who the high and low earners are.
Scale independence: the Gini coefficient does not consider the size of the economy, the way it is measured, or whether it is a rich or poor country on average.
Population independence: it does not matter how large the population of the country is.
Transfer principle: if income (less than the difference), is transferred from a rich person to a poor person the resulting distribution is more equal.
Countries by Gini Index
A Gini index value above 50 is considered high; countries including Brazil, Colombia, South Africa, Botswana, and Honduras can be found in this category. A Gini index value of 30 or above is considered medium; countries including Vietnam, Mexico, Poland, The United States, Argentina, Russia and Uruguay can be found in this category. A Gini index value lower than 30 is considered low; countries including Austria, Germany, Denmark, Slovenia, Sweden and Ukraine can be found in this category.
Limitations of Gini coefficient
The Gini coefficient is a relative measure. Its proper use and interpretation is controversial. It is possible for the Gini coefficient of a developing country to rise (due to increasing inequality of income) while the number of people in absolute poverty decreases. This is because the Gini coefficient measures relative, not absolute, wealth. Changing income inequality, measured by Gini coefficients, can be due to structural changes in a society such as growing population (baby booms, aging populations, increased divorce rates, extended family households splitting into nuclear families, emigration, immigration) and income mobility. Gini coefficients are simple, and this simplicity can lead to oversights and can confuse the comparison of different populations; for example, while both Bangladesh (per capita income of $1,693) and the Netherlands (per capita income of $42,183) had an income Gini coefficient of 0.31 in 2010, the quality of life, economic opportunity and absolute income in these countries are very different, i.e. countries may have identical Gini coefficients, but differ greatly in wealth. Basic necessities may be available to all in a developed economy, while in an undeveloped economy with the same Gini coefficient, basic necessities may be unavailable to most or unequally available, due to lower absolute wealth.
- Different income distributions with the same Gini coefficient
Even when the total income of a population is the same, in certain situations two countries with different income distributions can have the same Gini index (e.g. cases when income Lorenz Curves cross). Table A illustrates one such situation. Both countries have a Gini coefficient of 0.2, but the average income distributions for household groups are different. As another example, in a population where the lowest 50% of individuals have no income and the other 50% have equal income, the Gini coefficient is 0.5; whereas for another population where the lowest 75% of people have 25% of income and the top 25% have 75% of the income, the Gini index is also 0.5. Economies with similar incomes and Gini coefficients can have very different income distributions. Bellù and Liberati claim that to rank income inequality between two different populations based on their Gini indices is sometimes not possible, or misleading.
- Extreme wealth inequality, yet low income Gini coefficient
A Gini index does not contain information about absolute national or personal incomes. Populations can have very low income Gini indices, yet simultaneously very high wealth Gini index. By measuring inequality in income, the Gini ignores the differential efficiency of use of household income. By ignoring wealth (except as it contributes to income) the Gini can create the appearance of inequality when the people compared are at different stages in their life. Wealthy countries such as Sweden can show a low Gini coefficient for disposable income of 0.31 thereby appearing equal, yet have very high Gini coefficient for wealth of 0.79 to 0.86 thereby suggesting an extremely unequal wealth distribution in its society. These factors are not assessed in income-based Gini.
|1||20,000||1 & 2||50,000|
|3||40,000||3 & 4||90,000|
|5||60,000||5 & 6||130,000|
|7||80,000||7 & 8||170,000|
|9||120,000||9 & 10||270,000|
- Small sample bias – sparsely populated regions more likely to have low Gini coefficient
Gini index has a downward-bias for small populations. Counties or states or countries with small populations and less diverse economies will tend to report small Gini coefficients. For economically diverse large population groups, a much higher coefficient is expected than for each of its regions. Taking world economy as one, and income distribution for all human beings, for example, different scholars estimate global Gini index to range between 0.61 and 0.68. As with other inequality coefficients, the Gini coefficient is influenced by the granularity of the measurements. For example, five 20% quantiles (low granularity) will usually yield a lower Gini coefficient than twenty 5% quantiles (high granularity) for the same distribution. Philippe Monfort has shown that using inconsistent or unspecified granularity limits the usefulness of Gini coefficient measurements.
The Gini coefficient measure gives different results when applied to individuals instead of households, for the same economy and same income distributions. If household data is used, the measured value of income Gini depends on how the household is defined. When different populations are not measured with consistent definitions, comparison is not meaningful.
Deininger and Squire (1996) show that income Gini coefficient based on individual income, rather than household income, are different. For example, for the United States, they find that the individual income-based Gini index was 0.35, while for France it was 0.43. According to their individual focused method, in the 108 countries they studied, South Africa had the world's highest Gini coefficient at 0.62, Malaysia had Asia's highest Gini coefficient at 0.5, Brazil the highest at 0.57 in Latin America and Caribbean region, and Turkey the highest at 0.5 in OECD countries.
(in 2010 adjusted dollars)
|% of Population|
|% of Population|
|$15,000 – $24,999||11.9%||12.0%|
|$25,000 – $34,999||12.1%||10.9%|
|$35,000 – $49,999||15.4%||13.9%|
|$50,000 – $74,999||22.1%||17.7%|
|$75,000 – $99,999||12.4%||11.4%|
|$100,000 – $149,999||8.3%||12.1%|
|$150,000 – $199,999||2.0%||4.5%|
|$200,000 and over||1.2%||3.9%|
|United States' Gini|
on pre-tax basis
- Gini coefficient is unable to discern the effects of structural changes in populations
Expanding on the importance of life-span measures, the Gini coefficient as a point-estimate of equality at a certain time, ignores life-span changes in income. Typically, increases in the proportion of young or old members of a society will drive apparent changes in equality, simply because people generally have lower incomes and wealth when they are young than when they are old. Because of this, factors such as age distribution within a population and mobility within income classes can create the appearance of inequality when none exist taking into account demographic effects. Thus a given economy may have a higher Gini coefficient at any one point in time compared to another, while the Gini coefficient calculated over individuals' lifetime income is actually lower than the apparently more equal (at a given point in time) economy's. Essentially, what matters is not just inequality in any particular year, but the composition of the distribution over time.
Kwok claims income Gini coefficient for Hong Kong has been high (0.434 in 2010), in part because of structural changes in its population. Over recent decades, Hong Kong has witnessed increasing numbers of small households, elderly households and elderly living alone. The combined income is now split into more households. Many old people are living separately from their children in Hong Kong. These social changes have caused substantial changes in household income distribution. Income Gini coefficient, claims Kwok, does not discern these structural changes in its society. Household money income distribution for the United States, summarized in Table C of this section, confirms that this issue is not limited to just Hong Kong. According to the US Census Bureau, between 1979 and 2010, the population of United States experienced structural changes in overall households, the income for all income brackets increased in inflation-adjusted terms, household income distributions shifted into higher income brackets over time, while the income Gini coefficient increased.
Another limitation of Gini coefficient is that it is not a proper measure of egalitarianism, as it is only measures income dispersion. For example, if two equally egalitarian countries pursue different immigration policies, the country accepting a higher proportion of low-income or impoverished migrants will report a higher Gini coefficient and therefore may appear to exhibit more income inequality.
- Inability to value benefits and income frominformal economyaffects Gini coefficient accuracy
Some countries distribute benefits that are difficult to value. Countries that provide subsidized housing, medical care, education or other such services are difficult to value objectively, as it depends on quality and extent of the benefit. In absence of free markets, valuing these income transfers as household income is subjective. The theoretical model of Gini coefficient is limited to accepting correct or incorrect subjective assumptions.
In subsistence-driven and informal economies, people may have significant income in other forms than money, for example through subsistence farming or bartering. These income tend to accrue to the segment of population that is below-poverty line or very poor, in emerging and transitional economy countries such as those in sub-Saharan Africa, Latin America, Asia and Eastern Europe. Informal economy accounts for over half of global employment and as much as 90 per cent of employment in some of the poorer sub-Saharan countries with high official Gini inequality coefficients. Schneider et al., in their 2010 study of 162 countries, report about 31.2%, or about $20 trillion, of world's GDP is informal. In developing countries, the informal economy predominates for all income brackets except for the richer, urban upper income bracket populations. Even in developed economies, between 8% (United States) to 27% (Italy) of each nation's GDP is informal, and resulting informal income predominates as a livelihood activity for those in the lowest income brackets. The value and distribution of the incomes from informal or underground economy is difficult to quantify, making true income Gini coefficients estimates difficult. Different assumptions and quantifications of these incomes will yield different Gini coefficients.
Gini has some mathematical limitations as well. It is not additive and different sets of people cannot be averaged to obtain the Gini coefficient of all the people in the sets.
Alternatives to Gini coefficient
Given the limitations of Gini coefficient, other statistical methods are used in combination or as an alternative measure of population dispersity. For example, entropy measures are frequently used (e.g. the Atkinson index or the Theil Index and Mean log deviation as special cases of the generalized entropy index). These measures attempt to compare the distribution of resources by intelligent agents in the market with a maximum entropy random distribution, which would occur if these agents acted like non-interacting particles in a closed system following the laws of statistical physics.
Relation to other statistical measures
Although the Gini coefficient is most popular in economics, it can in theory be applied in any field of science that studies a distribution. For example, in ecology the Gini coefficient has been used as a measure of biodiversity, where the cumulative proportion of species is plotted against cumulative proportion of individuals. In health, it has been used as a measure of the inequality of health related quality of life in a population. In education, it has been used as a measure of the inequality of universities. In chemistry it has been used to express the selectivity of protein kinase inhibitors against a panel of kinases. In engineering, it has been used to evaluate the fairness achieved by Internet routers in scheduling packet transmissions from different flows of traffic.
The Gini coefficient is sometimes used for the measurement of the discriminatory power of rating systems in credit risk management.
A 2005 study accessed US census data to measure home computer ownership, and used the Gini coefficient to measure inequalities amongst whites and African Americans. Results indicated that although decreasing overall, home computer ownership inequality is substantially smaller among white households.
A 2016 peer-reviewed study titled Employing the Gini coefficient to measure participation inequality in treatment-focused Digital Health Social Networks illustrated that the Gini coefficient was helpful and accurate in measuring shifts in inequality, however as a standalone metric it failed to incorporate overall network size.
Great Gatsby curve
Human Poverty Index
Income inequality metrics
Hoover index (a.k.a. Robin Hood index)
Social welfare provision
List of countries by distribution of wealth
List of countries by income equality
List of U.S. states by Gini coefficient