Kurtosis standard error

The standard errors are valid for normal distributions, but not for other distributions. To see why, you can run the following code (which uses the spssSkewKurtosis function shown above) to estimate the true confidence level of the interval obtained by taking the kurtosis estimate plus or minus 1.96 standard errors:

set.seed(12345)
Nsim = 10000
Correct = numeric(Nsim)
b1.ols = numeric(Nsim)
b1.alt = numeric(Nsim)
for (i in 1:Nsim) {
 Data = rnorm(1000)  
 Kurt = spssSkewKurtosis(Data)[2,1]
 seKurt =  spssSkewKurtosis(Data)[2,2]
  LowerLimit = Kurt -1.96*seKurt
  UpperLimit = Kurt +1.96*seKurt
  Correct[i] = LowerLimit <= 0 & 0 <= UpperLimit  
 }

TrueConfLevel = mean(Correct)
TrueConfLevel

This gives you 0.9496, acceptably close to the expected 95%, so the standard errors work as expected when the data come from a normal distribution. But if you change Data = rnorm(1000) to Data = runif(1000), then you are assuming that the data come from a uniform distribution, whose theoretical (excess) kurtosis is -1.2. Making the corresponding change from Correct[i] = LowerLimit <= 0 & 0 <= UpperLimit to Correct[i] = LowerLimit <= -1.2 & -1.2 <= UpperLimit gives the result 1.0, meaning that the 95% intervals were always correct, rather than correct for 95% of the samples. Hence, the standard error seems to be overestimated (too large) for the (light-tailed) uniform distribution.

If you change Data = rnorm(1000) to Data = rexp(1000), then you are assuming that the data come from an exponential distribution, whose theoretical (excess) kurtosis is 6.0. Making the corresponding change from Correct[i] = LowerLimit <= 0 & 0 <= UpperLimit to Correct[i] = LowerLimit <= 6.0 & 6.0 <= UpperLimit gives the result 0.1007, meaning that the 95% intervals were correct only for 10.07% of the samples, rather than correct for 95% of the samples. Hence, the standard error seems to be underestimated (too small) for the (heavy-tailed) exponential distribution.

Those standard errors are grossly incorrect for non-normal distributions, as the simulation above shows. Thus, the only use of those standard errors is to compare the estimated kurtosis with the expected theoretical normal value (0.0); e.g., using a test of hypothesis. They cannot be used to construct a confidence interval for the true kurtosis.

In probability theory and statistics, kurtosis (from Greek: κυρτός, kyrtos or kurtos, meaning «curved, arching») is a measure of the «tailedness» of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes a particular aspect of a probability distribution. There are different ways to quantify kurtosis for a theoretical distribution, and there are corresponding ways of estimating it using a sample from a population. Different measures of kurtosis may have different interpretations.

The standard measure of a distribution’s kurtosis, originating with Karl Pearson,^[1] is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak;^[2] hence, the sometimes-seen characterization of kurtosis as «peakedness» is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean.

It is common to compare the excess kurtosis (defined below) of a distribution to 0, which is the excess kurtosis of any univariate normal distribution. Distributions with negative excess kurtosis are said to be platykurtic, although this does not imply the distribution is «flat-topped» as is sometimes stated. Rather, it means the distribution produces fewer and/or less extreme outliers than the normal distribution. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with a positive excess kurtosis are said to be leptokurtic. An example of a leptokurtic distribution is the Laplace distribution, which has tails that asymptotically approach zero more slowly than a Gaussian, and therefore produces more outliers than the normal distribution. It is common practice to use excess kurtosis, which is defined as Pearson’s kurtosis minus 3, to provide a simple comparison to the normal distribution. Some authors and software packages use «kurtosis» by itself to refer to the excess kurtosis. For clarity and generality, however, this article explicitly indicates where non-excess kurtosis is meant.

Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on four population or sample quantiles.^[3] These are analogous to the alternative measures of skewness that are not based on ordinary moments.^[3]

Pearson moments[edit]

The kurtosis is the fourth standardized moment, defined as

where μ₄ is the fourth central moment and σ is the standard deviation. Several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ₂, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis.

The kurtosis is bounded below by the squared skewness plus 1:^[4]: 432

where μ₃ is the third central moment. The lower bound is realized by the Bernoulli distribution. There is no upper limit to the kurtosis of a general probability distribution, and it may be infinite.

A reason why some authors favor the excess kurtosis is that cumulants are extensive. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let X₁, …, X_n be independent random variables for which the fourth moment exists, and let Y be the random variable defined by the sum of the X_i. The excess kurtosis of Y is

where is the standard deviation of . In particular if all of the X_i have the same variance, then this simplifies to

The reason not to subtract 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed. The cokurtosis between pairs of variables is an order four tensor. For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to «correct» for an excess becomes confusing. It is true, however, that the joint cumulants of degree greater than two for any multivariate normal distribution are zero.

For two random variables, X and Y, not necessarily independent, the kurtosis of the sum, X + Y, is

Note that the fourth-power binomial coefficients (1, 4, 6, 4, 1) appear in the above equation.

Interpretation[edit]

The exact interpretation of the Pearson measure of kurtosis (or excess kurtosis) used to be disputed, but is now settled. As Westfall notes in 2014^[2], «…its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution).» The logic is simple: Kurtosis is the average (or expected value) of the standardized data raised to the fourth power. Standardized values that are less than 1 (i.e., data within one standard deviation of the mean, where the «peak» would be) contribute virtually nothing to kurtosis, since raising a number that is less than 1 to the fourth power makes it closer to zero. The only data values (observed or observable) that contribute to kurtosis in any meaningful way are those outside the region of the peak; i.e., the outliers. Therefore, kurtosis measures outliers only; it measures nothing about the «peak».

Many incorrect interpretations of kurtosis that involve notions of peakedness have been given. One is that kurtosis measures both the «peakedness» of the distribution and the heaviness of its tail.^[5] Various other incorrect interpretations have been suggested, such as «lack of shoulders» (where the «shoulder» is defined vaguely as the area between the peak and the tail, or more specifically as the area about one standard deviation from the mean) or «bimodality».^[6] Balanda and MacGillivray assert that the standard definition of kurtosis «is a poor measure of the kurtosis, peakedness, or tail weight of a distribution»^[5]: 114 and instead propose to «define kurtosis vaguely as the location- and scale-free movement of probability mass from the shoulders of a distribution into its center and tails».^[5]

Moors’ interpretation[edit]

In 1986 Moors gave an interpretation of kurtosis.^[7] Let

where X is a random variable, μ is the mean and σ is the standard deviation.

Now by definition of the kurtosis , and by the well-known identity

.

The kurtosis can now be seen as a measure of the dispersion of Z² around its expectation. Alternatively it can be seen to be a measure of the dispersion of Z around +1 and −1. κ attains its minimal value in a symmetric two-point distribution. In terms of the original variable X, the kurtosis is a measure of the dispersion of X around the two values μ ± σ.

High values of κ arise in two circumstances:

• where the probability mass is concentrated around the mean and the data-generating process produces occasional values far from the mean,
• where the probability mass is concentrated in the tails of the distribution.

Excess kurtosis[edit]

The excess kurtosis is defined as kurtosis minus 3. There are 3 distinct regimes as described below.

Mesokurtic[edit]

Distributions with zero excess kurtosis are called mesokurtic, or mesokurtotic. The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of its parameters. A few other well-known distributions can be mesokurtic, depending on parameter values: for example, the binomial distribution is mesokurtic for .

Leptokurtic[edit]

A distribution with positive excess kurtosis is called leptokurtic, or leptokurtotic. «Lepto-» means «slender».^[8] In terms of shape, a leptokurtic distribution has fatter tails. Examples of leptokurtic distributions include the Student’s t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution. Such distributions are sometimes termed super-Gaussian.^[9]

Platykurtic[edit]

A distribution with negative excess kurtosis is called platykurtic, or platykurtotic. «Platy-» means «broad».^[10] In terms of shape, a platykurtic distribution has thinner tails. Examples of platykurtic distributions include the continuous and discrete uniform distributions, and the raised cosine distribution. The most platykurtic distribution of all is the Bernoulli distribution with p = 1/2 (for example the number of times one obtains «heads» when flipping a coin once, a coin toss), for which the excess kurtosis is −2.

Graphical examples[edit]

The Pearson type VII family[edit]

The effects of kurtosis are illustrated using a parametric family of distributions whose kurtosis can be adjusted while their lower-order moments and cumulants remain constant. Consider the Pearson type VII family, which is a special case of the Pearson type IV family restricted to symmetric densities. The probability density function is given by

where a is a scale parameter and m is a shape parameter.

All densities in this family are symmetric. The kth moment exists provided m > (k + 1)/2. For the kurtosis to exist, we require m > 5/2. Then the mean and skewness exist and are both identically zero. Setting a² = 2m − 3 makes the variance equal to unity. Then the only free parameter is m, which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize with , where is the excess kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary non-negative excess kurtosis. The reparameterized density is

In the limit as one obtains the density

which is shown as the red curve in the images on the right.

In the other direction as one obtains the standard normal density as the limiting distribution, shown as the black curve.

In the images on the right, the blue curve represents the density with excess kurtosis of 2. The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density, although this conclusion is only valid for this select family of distributions. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is a parabola. One can see that the normal density allocates little probability mass to the regions far from the mean («has thin tails»), compared with the blue curve of the leptokurtic Pearson type VII density with excess kurtosis of 2. Between the blue curve and the black are other Pearson type VII densities with γ₂ = 1, 1/2, 1/4, 1/8, and 1/16. The red curve again shows the upper limit of the Pearson type VII family, with (which, strictly speaking, means that the fourth moment does not exist). The red curve decreases the slowest as one moves outward from the origin («has fat tails»).

Other well-known distributions[edit]

Several well-known, unimodal, and symmetric distributions from different parametric families are compared here. Each has a mean and skewness of zero. The parameters have been chosen to result in a variance equal to 1 in each case. The images on the right show curves for the following seven densities, on a linear scale and logarithmic scale:

• D: Laplace distribution, also known as the double exponential distribution, red curve (two straight lines in the log-scale plot), excess kurtosis = 3
• S: hyperbolic secant distribution, orange curve, excess kurtosis = 2
• L: logistic distribution, green curve, excess kurtosis = 1.2
• N: normal distribution, black curve (inverted parabola in the log-scale plot), excess kurtosis = 0
• C: raised cosine distribution, cyan curve, excess kurtosis = −0.593762…
• W: Wigner semicircle distribution, blue curve, excess kurtosis = −1
• U: uniform distribution, magenta curve (shown for clarity as a rectangle in both images), excess kurtosis = −1.2.

Note that in these cases the platykurtic densities have bounded support, whereas the densities with positive or zero excess kurtosis are supported on the whole real line.

One cannot infer that high or low kurtosis distributions have the characteristics indicated by these examples. There exist platykurtic densities with infinite support,

• e.g., exponential power distributions with sufficiently large shape parameter b

and there exist leptokurtic densities with finite support.

• e.g., a distribution that is uniform between −3 and −0.3, between −0.3 and 0.3, and between 0.3 and 3, with the same density in the (−3, −0.3) and (0.3, 3) intervals, but with 20 times more density in the (−0.3, 0.3) interval

Also, there exist platykurtic densities with infinite peakedness,

• e.g., an equal mixture of the beta distribution with parameters 0.5 and 1 with its reflection about 0.0

and there exist leptokurtic densities that appear flat-topped,

• e.g., a mixture of distribution that is uniform between -1 and 1 with a T(4.0000001) Student’s t-distribution, with mixing probabilities 0.999 and 0.001.

Sample kurtosis[edit]

Definitions[edit]

A natural but biased estimator[edit]

For a sample of n values, a method of moments estimator of the population excess kurtosis can be defined as

where m₄ is the fourth sample moment about the mean, m₂ is the second sample moment about the mean (that is, the sample variance), x_i is the i^th value, and is the sample mean.

This formula has the simpler representation,

where the values are the standardized data values using the standard deviation defined using n rather than n − 1 in the denominator.

For example, suppose the data values are 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999.

Then the values are −0.239, −0.225, −0.221, −0.234, −0.230, −0.225, −0.239, −0.230, −0.234, −0.225, −0.230, −0.239, −0.230, −0.230, −0.225, −0.230, −0.216, −0.230, −0.225, 4.359

and the values are 0.003, 0.003, 0.002, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.002, 0.003, 0.003, 360.976.

The average of these values is 18.05 and the excess kurtosis is thus 18.05 − 3 = 15.05. This example makes it clear that data near the «middle» or «peak» of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measure «peakedness». It is simply a measure of the outlier, 999 in this example.

Standard unbiased estimator[edit]

Given a sub-set of samples from a population, the sample excess kurtosis above is a biased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis, which is unbiased in random samples of a normal distribution, is defined as follows:^[3]

where k₄ is the unique symmetric unbiased estimator of the fourth cumulant, k₂ is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), m₄ is the fourth sample moment about the mean, m₂ is the second sample moment about the mean, x_i is the i^th value, and is the sample mean. This adjusted Fisher–Pearson standardized moment coefficient is the version found in Excel and several statistical packages including Minitab, SAS, and SPSS.^[11]

Unfortunately, in nonnormal samples is itself generally biased.

Upper bound[edit]

An upper bound for the sample kurtosis of n (n > 2) real numbers is^[12]

where is the corresponding sample skewness.

Variance under normality[edit]

The variance of the sample kurtosis of a sample of size n from the normal distribution is^[13]

Stated differently, under the assumption that the underlying random variable is normally distributed, it can be shown that .^{[14]: Page number needed}

Applications[edit]

This section needs expansion. You can help by adding to it. (December 2009)

The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods.

D’Agostino’s K-squared test is a goodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is the Jarque–Bera test for normality.

For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance.

Pearson’s definition of kurtosis is used as an indicator of intermittency in turbulence.^[15] It is also used in magnetic resonance imaging to quantify non-Gaussian diffusion.^[16]

A concrete example is the following lemma by He, Zhang, and Zhang:^[17]
Assume a random variable has expectation , variance and kurtosis .
Assume we sample many independent copies. Then

.

This shows that with many samples, we will see one that is above the expectation with probability at least .
In other words: If the kurtosis is large, we might see a lot values either all below or above the mean.

Kurtosis convergence[edit]

Applying band-pass filters to digital images, kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termed kurtosis convergence, can be used to detect image splicing in forensic analysis.^[18]

Other measures[edit]

A different measure of «kurtosis» is provided by using L-moments instead of the ordinary moments.^[19][20]

See also[edit]

• Kurtosis risk
• Maximum entropy probability distribution

References[edit]

Further reading[edit]

• Kim, Tae-Hwan; White, Halbert (2003). «On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index». Finance Research Letters. 1: 56–70. doi:10.1016/S1544-6123(03)00003-5. S2CID 16913409. Alternative source (Comparison of kurtosis estimators)
• Seier, E.; Bonett, D.G. (2003). «Two families of kurtosis measures». Metrika. 58: 59–70. doi:10.1007/s001840200223. S2CID 115990880.

External links[edit]

• «Excess coefficient», Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Kurtosis calculator
• Free Online Software (Calculator) computes various types of skewness and kurtosis statistics for any dataset (includes small and large sample tests)..
• Kurtosis on the Earliest known uses of some of the words of mathematics
• Celebrating 100 years of Kurtosis a history of the topic, with different measures of kurtosis.

Mean — средняя
арифметическая;

Среднее значение
случайной величины представляет собой
наиболее типичное, наиболее вероятное
ее значение, своеобразный центр, вокруг
которого разбросаны все значения
признака.

Sum — сумма;

Median — медиана;

Медианой является
такое значение случайной величины,
которое разделяет все случаи выборки
на две равные по численности части.

Standard Deviation
— стандартное отклонение;

Стандартное
отклонение (или среднее квадратическое
отклонение) является мерой изменчивости
(вариации) признака. Оно показывает на
какую величину в среднем отклоняются
случаи от среднего значения признака.
Особенно большое значение имеет при
исследовании нормальных распределений.
В нормальном распределении 68% всех
случаев лежит в интервале +
одного отклонения от среднего, 95% — +
двух стандартных отклонений от среднего
и 99,7% всех случаев — в интервале +
трех стандартных отклонений от среднего.

V Рис. 8. Окно выбора статистик ariance — дисперсия;

Дисперсия является
мерой изменчивости, вариации признака
и представляет собой средний квадрат
отклонений случаев от среднего значения
признака. В отличии от других показателей
вариации дисперсия может быть разложена
на составные части, что позволяет тем
самым оценить влияние различных факторов
на вариацию признака. Дисперсия — один
из существеннейших показателей,
характеризующих явление или процесс,
один из основных критериев возможности
создания достаточно точных моделей.

Standard error of mean
— стандартная ошибка
среднего;

Стандартная ошибка
среднего это величина, на которую
отличается среднее значение выборки
от среднего значения генеральной
совокупности при условии, что распределение
близко к нормальному. С вероятностью
0,68 можно утверждать, что среднее значение
генеральной совокупности лежит в
интервале +
одной стандартной ошибки от среднего,
с вероятностью 0,95 — в интервале +
двух стандартных ошибок от среднего и
с вероятностью 0,99 — среднее значение
генеральной совокупности лежит в
интервале +
трех стандартных ошибок от среднего.

95% Confidence limits of mean — 95%-ый доверительный интервал для среднего;

Интервал, в который
с вероятностью 0,95 попадает среднее
значение признака генеральной
совокупности.

Minimum, maximum
— минимальное и максимальное значения;

Lower, upper
quartiles — нижний
и верхний квартили;

Верхний квартиль
это такое значение случайной величины,
больше которого по величине 25% случаев
выборки. Верхний квартиль это такое
значение случайной величины, меньше
которого по величине 25% случаев выборки.

Range — размах;

Расстояние между
наибольшим (maximum)
и наименьшим (minimum)
значениями признака.

Quartile range
— интерквартильная широта;

Расстояние между
нижним и верхним квартилями.

Skewness -асимметрия;

Асимметрия
характеризует степень смещения
вариационного ряда относительно среднего
значения по величине и направлению. В
симметричной кривой коэффициент
асимметрии равен нулю. Если правая ветвь
кривой, начиная от вершины) больше левой
(правосторонняя асимметрия), то коэффициент
асимметрии больше нуля. Если левая ветвь
кривой больше правой (левосторонняя
асимметрия), то коэффициент асимметрии
меньше нуля. Асимметрия менее 0,5 считается
малой.

Standard error
of Skewness
-стандартная ошибка асимметрии;

Kurtosis — эксцесс;

Эксцесс характеризует
степень концентрации случаев вокруг
среднего значения и является своеобразной
мерой крутости кривой. В кривой нормального
распределения эксцесс равен нулю. Если
эксцесс больше нуля, то кривая распределения
характеризуется островершинностью,
т.е. является более крутой по сравнению
с нормальной, а случаи более густо
группируются вокруг среднего. При
отрицательном эксцессе кривая является
более плосковершинной, т.е. более пологой
по сравнению с нормальным распределением.
Отрицательным пределом величины эксцесса
является число -2, положительного предела
— нет.

Standard error
of Kurtosis
— стандартная ошибка эксцесса.

На против статистик, подлежащих вычислению
(рис. следует поставить флажок.

После нажатия на кнопку OK
окна Descriptive statistics
на экране появится таблица с результатами
расчетов описательных статистик (рис.
9).

Рис. 9. Окно с результатами расчета
описательных статистик

В таблице 2 эти данные представлены
после копирования в текстовый редактор
Word.

К сожалению, пакет Statistica
не рассчитывает такие часто применяемые
статистики, как коэффициент вариации
и относительная ошибка среднего значения
(точность опыта). Но их определение не
представляет большого труда. Коэффициент
вариации (%) есть отношение стандартного
отклонения к среднему значению, умноженное
на 100%:

Коэффициент
вариации, как дисперсия и стандартное
отклонение, является показателем
изменчивости признака. Коэффициент
вариации не зависит от единиц измерения,
поэтому удобен для сравнительной оценки
различных статистических совокупностей.
При величине коэффициента вариации до
10% изменчивость оценивается как слабая,
11-25% — средняя, более 25% — сильная (Лакин,
1990).

О
тносительная
ошибка среднего значения (%) — отношение
стандартной ошибки среднего к среднему
значению, умноженное на 100% (для вероятности
0,68):

Это процент
расхождения между генеральной и
выборочной средней, показывает на
сколько процентов можно ошибиться, если
утверждать, что генеральная средняя
равна выборочной средней. Если
относительная ошибка не превышает 5%,
то точность исследований (точность
опыта) оценивается как хорошая, до 10% —
удовлетворительная.

Точность 3-5% при
вероятности 0,95, а в некоторых случаях
и при вероятности 0,68, является вполне
достаточной для большинства задач
лесного хозяйства.

П

Рис.106. Окно задания

переменной- весов

ри необходимости обработки
сгруппированных данных нужно
воспользоваться кнопкой Weight
окна Descriptive statistics
(рис.4). В появляющемся диалоговом окне
(рис. 10) следует указать переменную,
являющуюся весами для других переменных
(Weight variables),
а переключатель Status
установить в положение ON.
Необходимо иметь в виду, что весы
действуют сразу для всех переменных.
Поэтому обрабатывать сгруппированные
и не сгруппированные данные нужно
отдельно.

При помощи опции Alpha
error (рис. 4)
выбирается уровень доверительной
вероятности статистического анализа.
В биологических исследованиях наиболее
часто используется вероятность 0,95
(95%). Вероятности 0,95 соответствует уровень
значимости 0,05 (5%).

Кнопка Select cases
позволяет установить условия включения
(include if) или
исключения (exclude if)
случаев (строк файла данных) из
статистической обработки (рис. 11).
Операторы, которые могут использоваться
при написании выражений, а также примеры
самих выражений имеются непосредственно
на самом диалоговом окне Case
Selection Conditions
(рис. 11) в нижней его части.

Рис. 11. Окно задания условий выбора
случаев

Для визуализации описательных статистик
можно построить статистические графики
типа «коробок» (или «ящиков с
усами»). Это легко можно сделать при
помощи кнопки Box &
Whisker plot
for all
variable окна Descriptive
statistics. На графике можно
отобразить 3 статистики, установив
переключатель в одно из 4-х положений
(рис. 12):

Рис. 12. Окно выбора статистик для
графика коробок

1. Median/Quart./Range
   — Медиана / Квартили / Размах;

2. Mean/SE/SD
   — Среднее / Ошибка среднего / Стандартное
   отклонение

3. Mean/SD/1.96SD
   — Среднее / Стандартное отклонение /
   Интервал 1,96* стандартного отклонения;

4. Mean/SE/1.96*SE
   — Среднее / Ошибка среднего / Интервал
   1,96 * ошибки среднего.

В
изуализация
описательных статистик переменных
VAR1, VAR3 и
VAR4 рассматриваемого
примера при помощи графика коробок
представлена на рис. 13.

Рис. 13. Описательные статистики в
графическом виде

Соседние файлы в папке Статистика

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