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Learning Outcomes

    Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.

Nội dung chính

    Learning OutcomesStep 1: Calculate the expected frequenciesStep 2: Calculate chi-squareStep 3: Find the critical chi-square valueStep 4: Compare the chi-square value to the critical valueStep 5: Decide whether the reject the null hypothesisWhich measure of central tendency is most affected by extreme very high or very low scores?Which measure of central tendency is most affected by an outlier?Which measure of central tendency is most affected?

By now, everyone should know how to calculate mean, median and mode. They each give us a measure of Central Tendency (i.e. where the center of our data falls), but often give different answers. So how do we know when to use each? Here are some general rules:

 Mean is the most
frequently used measure of central tendency and generally considered the best measure of it. However, there are some situations where either median or mode are preferred.Median is the preferred measure of central tendency when:  There are a few extreme scores in the distribution of the data. (NOTE: Remember that a single outlier can have a great effect on the mean). b.There are some missing or undetermined values in your data. c.There is an open ended
distribution (For example, if you have a data field which measures number of children and your options are [latex]0[/latex], [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], [latex]5[/latex] or “[latex]6[/latex] or more,” than the “[latex]6[/latex] or more field” is open ended and makes calculating the mean impossible, since we do not know exact values for this field).You have data measured on an ordinal scale.Mode is the preferred measure when
data are measured in a nominal ( and even sometimes ordinal) scale.

What’s the difference between relative frequency and probability?

Probability is the relative frequency over an infinite number of trials.

For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.

Since doing something an infinite number of times
is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.

How do I perform a chi-square goodness of fit test for a genetic cross?

Chi-square goodness of fit tests are often used in genetics. One common application is to check if two genes are linked (i.e., if the assortment is independent). When genes are linked, the allele inherited for one gene affects the allele inherited for another gene.

Suppose that you want to know if the genes for pea texture (R = round, r = wrinkled) and color
(Y = yellow, y = green) are linked. You perform a dihybrid cross between two heterozygous (RY / ry) pea plants. The hypotheses you’re testing with your experiment are:

    Null hypothesis (H0): The population of offspring have an equal probability of inheriting all possible genotypic combinations.

      This would suggest that the genes are unlinked.

    Alternative hypothesis (Ha): The population of offspring do not have an equal probability of inheriting all possible genotypic combinations.

      This would suggest that the genes are linked.

You observe 100 peas:

    78 round and yellow peas6 round and green peas4 wrinkled and yellow peas12 wrinkled and green peas

Step 1: Calculate the expected frequencies

To calculate the expected values, you can make a Punnett square. If the two genes are unlinked, the probability of each genotypic combination is equal.

RY
ry
Ry
rY
RY
RRYY
RrYy
RRYy
RrYY
ry
RrYy
rryy
Rryy
rrYy
Ry
RRYy
Rryy
RRyy
RrYy
rY
RrYY
rrYy
RrYy
rrYY

The expected phenotypic ratios are therefore 9 round and yellow: 3 round and green: 3 wrinkled and yellow: 1 wrinkled and green.

From this, you can calculate the expected phenotypic frequencies for 100 peas:

Phenotype
Observed
Expected
Round and yellow
78
100 * (9/16) = 56.25
Round and green
6
100 * (3/16) = 18.75
Wrinkled and yellow
4
100 * (3/16) = 18.75
Wrinkled and green
12
100 * (1/16) = 6.21

Step 2: Calculate chi-square

Phenotype
Observed
Expected
OE
(O − E)2
(O − E)2 / E
Round and yellow
78
56.25
21.75
473.06
8.41
Round and green
6
18.75
−12.75
162.56
8.67
Wrinkled and yellow
4
18.75
−14.75
217.56
11.6
Wrinkled and green
12
6.21
5.79
33.52
5.4

Χ2 = 8.41 + 8.67 + 11.6 + 5.4 = 34.08

Step 3: Find the critical chi-square value

Since there are four groups (round and yellow, round and green, wrinkled and yellow, wrinkled and green), there are three degrees of freedom.

For a test of significance α = .05 and df = 3, the Χ2 critical value is 7.82.

Step 4: Compare the chi-square value to the critical value

Χ2 = 34.08

Critical value = 7.82

The Χ2 value is greater than the critical value.

Step 5: Decide whether the reject the null hypothesis

The Χ2 value is greater than the critical value, so we reject the null hypothesis that the population of offspring have an equal probability of inheriting all possible genotypic combinations. There is a
significant difference between the observed and expected genotypic frequencies (p. < .05).

The data supports the alternative hypothesis that the offspring do not have an equal probability of inheriting all possible genotypic combinations, which suggests that the genes are linked

How do I find quartiles in R?

You can use the quantile() function to find quartiles in R. If
your data is called “data”, then “quantile(data, prob=c(.25,.5,.75), type=1)” will return the three quartiles.

How do I find quartiles in Excel?

You can use the QUARTILE() function to find quartiles in Excel. If your data is in column A, then click any blank cell and type “=QUARTILE(A:A,1)” for the first quartile, “=QUARTILE(A:A,2)” for the second quartile, and “=QUARTILE(A:A,3)” for the third quartile.

What are the three types of skewness?

The three types of skewness are:

    Right skew (also called positive skew). A right-skewed distribution is longer on the right side of its peak than on its left.Left skew (also called negative skew). A left-skewed distribution is longer on the left side of its peak than on its right.Zero
    skew.
    It is symmetrical and its left and right sides are mirror images.

How do I find the critical value of t in Excel?

You can use
the T.INV() function to find the critical value of t for one-tailed tests in Excel, and you can use the T.INV.2T() function for two-tailed tests.

Example: Calculating the critical value of t in ExcelTo calculate the critical value of t for a two-tailed test with df = 29 and α = .05, click any blank cell and type:

=T.INV.2T(0.05,29)

How do I find the critical value of t in R?

You can use the qt()
function to find the critical value of t in R. The function gives the critical value of t for the one-tailed test. If you want the critical value of t for a two-tailed test, divide the significance level by two.

Example: Calculating the critical value of t in RTo calculate the
critical value of t for a two-tailed test with df = 29 and α = .05:

qt(p. = .025, df = 29)

What are the types of missing data?

There are three main types of missing data.

Missing completely random (MCAR) data are randomly distributed across the variable and unrelated to other
variables.

Missing random (MAR) data are not randomly distributed but they are accounted for by other observed variables.

Missing not random (MNAR) data systematically differ from the observed values.

How do I giảm giá with missing data?

To tidy up your missing data, your options
usually include accepting, removing, or recreating the missing data.

    Acceptance: You leave your data as isListwise or pairwise deletion: You delete all cases (participants) with missing data from analysesImputation: You use other data to fill in the missing data

How do I calculate the geometric mean?

There are two steps to calculating the
geometric mean:

Multiply all values together to get their product.Find the nth root of the product (n is the number of values).

Before calculating the geometric mean, note that:

    The geometric mean can only be found for positive values.If any value in the data set
    is zero, the geometric mean is zero.

What’s the difference between the arithmetic and geometric means?

The arithmetic mean is the most commonly used type of mean and is often referred to simply as “the mean.” While the arithmetic mean is based on adding and dividing values, the
geometric mean multiplies and finds the root of values.

Even though the geometric mean is a less common measure of central tendency, it’s more accurate than the arithmetic mean for percentage change and positively skewed data. The geometric mean is often
reported for financial indices and population growth rates.

What are outliers?

Outliers are extreme values that differ from most values in the dataset. You find outliers the extreme ends of your dataset.

How
do I find outliers in my data?

You can choose from four main ways to detect outliers:

    Sorting your values from low to high and checking minimum and maximum valuesVisualizing your data with a box plot
    and looking for outliersUsing the interquartile range to create fences for your dataUsing statistical procedures to identify extreme values

What do the sign and value of the correlation coefficient tell you?

Correlation coefficients always range between -1 and 1.

The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.

The absolute value of a number is equal to the number without its sign. The absolute value of a
correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation.

How do you increase statistical power?

There are various ways to improve power:

    Increase the potential effect size by manipulating your independent variable more strongly,Increase sample size,Increase the
    significance level (alpha),Reduce measurement error by increasing the precision and accuracy of your measurement devices and procedures,Use a one-tailed test instead of a two-tailed test for t tests and z tests.

What is a power analysis?

A power analysis is a calculation that helps
you determine a minimum sample size for your study. It’s made up of four main components. If you know or have estimates for any three of these, you can calculate the fourth component.

    Statistical power: the likelihood that a test will detect an effect of a certain size if there is one, usually set 80% or higher.Sample size: the
    minimum number of observations needed to observe an effect of a certain size with a given power level.Significance level (alpha): the maximum risk of rejecting a true null hypothesis that you are willing to take, usually set 5%.Expected
    effect size
    : a standardized way of expressing the magnitude of the expected result of your study, usually based on similar studies or a pilot study.

How do you reduce the risk of making a Type I error?

The risk of making a
Type I error is the significance level (or alpha) that you choose. That’s a value that you set the beginning of your study to assess the statistical probability of obtaining your results
(p. value).

The significance level is usually set 0.05 or 5%. This means that your results only have a 5% chance of occurring, or less, if the null hypothesis is actually true.

To reduce the Type I error probability, you can set a lower significance level.

What is statistical power?

In statistics, power refers to the
likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative (a Type II error).

If you don’t ensure enough power in your study, you may not be able to detect a statistically
significant result even when it has practical significance. Your study might not have the ability to answer your research question.

How do I calculate effect size?

There are dozens of measures of effect sizes. The most common effect sizes are Cohen’s d and Pearson’s r. Cohen’s d measures the size of the difference between two groups while Pearson’s r measures the strength of the relationship between two
variables.

What is effect size?

Effect size tells you how meaningful the relationship between variables or the difference between groups is.

A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications.

What is standard error?

The standard error of the
mean, or simply standard error, indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to
repeat a study using new samples from within a single population.

How do you know whether a number is a parameter or a statistic?

To figure out whether a given number is a parameter or a statistic, ask yourself the following:

    Does the number describe a whole, complete population where every
    thành viên can be reached for data collection?Is it possible to collect data for this number from every thành viên of the population in a reasonable time frame?

If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters.

If the answer is no to either
of the questions, then the number is more likely to be a statistic.

What are the different types of means?

The arithmetic mean is the most commonly used mean. It’s often simply called the mean or the average. But there are some other types of means you can calculate depending on your research purposes:

    Weighted mean: some values contribute more to the mean than others.Geometric mean: values are multiplied rather than summed up.Harmonic mean: reciprocals of values are used instead of the values themselves.

How do I find the mean?

You can find the mean, or average, of a data set in two simple steps:

    Find the sum of the values by adding them all up.Divide the sum by the number of values in the data set.

This method is the same whether you are dealing with sample or population data or positive or negative numbers.

When should I use the median?

The median is the most informative measure of
central tendency for skewed distributions or distributions with outliers. For example, the median is often used as a measure of central tendency for income distributions, which are generally highly skewed.

Because the median only uses one or two values, it’s unaffected by extreme outliers or non-symmetric distributions of scores. In contrast, the
mean and mode can vary in skewed distributions.

Can there be more than one mode?

A data set can often have no mode, one mode or more than one mode – it all depends on how many different values repeat most frequently.

Your
data can be:

    without any modeunimodal, with one mode,bimodal, with two modes,trimodal, with three modes, ormultimodal, with four or more modes.

How do I find the mode?

To find the mode:

    If your data is numerical or quantitative, order the values from low to high.If it is categorical, sort the values by group, in any order.

Then you simply need to identify the most frequently occurring
value.

What are the two main methods for calculating interquartile range?

The two most common methods for calculating interquartile range are the exclusive and inclusive methods.

The exclusive method excludes the median when identifying Q1 and Q3, while the inclusive method includes the median as a value in the data set in identifying the quartiles.

For each of
these methods, you’ll need different procedures for finding the median, Q1 and Q3 depending on whether your sample size is even- or odd-numbered. The exclusive method works best for even-numbered sample sizes, while the inclusive method is often used with odd-numbered sample sizes.

What is homoscedasticity?

Homoscedasticity, or homogeneity of variances, is an assumption of equal or
similar variances in different groups being compared.

This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. Uneven variances in samples result in biased and skewed test results.

What is the empirical rule?

The empirical rule, or the 68-95-99.7 rule, tells you where
most of the values lie in a normal distribution:

    Around 68% of values are within 1 standard deviation of the mean.Around 95% of values are within 2 standard deviations of the mean.Around 99.7% of values are within 3 standard deviations of the
    mean.

The empirical rule is a quick way to get an overview of your data and check for any outliers or extreme values that don’t follow this pattern.

What does standard deviation
tell you?

The standard deviation is the average amount of variability in your data set. It tells you, on average, how far each score lies from
the mean.

In normal distributions, a high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the mean.

What is variability?

Variability tells you how far apart points lie from each other and from the
center of a distribution or a data set.

Variability is also referred to as spread, scatter or dispersion.

What is the difference between interval and ratio data?

While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero.

For
example, temperature in Celsius or Fahrenheit is an interval scale because zero is not the lowest possible temperature. In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy.

What is a
critical value?

A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence
interval, or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data (i.e. 90%, 95%, 99%).

If you are constructing a 95% confidence interval and are using a threshold of statistical significance of p. = 0.05,
then your critical value will be identical in both cases.

What is a t-score?

A
t-score (a.k.a. a t-value) is equivalent to the number of standard deviations away from the mean of the t-distribution.

The t-score is the test statistic used in
t-tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t-distribution.

What is a t-distribution?

The t-distribution
is a way of describing a set of observations where most observations fall close to the mean, and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the
variance in the data is unknown.

The t-distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation.

What is ordinal data?

Ordinal data has two characteristics:

    The data can be classified into different categories within a variable.The categories have a natural ranked order.

However, unlike with interval data, the distances between the categories are uneven or unknown.

What is nominal data?

Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable. These categories cannot be ordered in a meaningful way.

For example, for the nominal
variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.

What does it mean if my confidence interval includes zero?

If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups.

If your confidence
interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data.

In both of these cases, you will also find a high p.-value when you run
your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.

What are z-scores and t-scores?

The z-score and t-score (aka z-value and t-value) show how many
standard deviations away from the mean of the distribution you are, assuming your data follow a z-distribution or a t-distribution.

These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z-score of 2.5, this means that your estimate is 2.5 standard deviations from the predicted mean.

The predicted mean and distribution of your estimate are generated by the
null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.

What is the difference between a confidence interval and a confidence level?

The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way.

The confidence intervalconsists of the upper and lower bounds of the estimate you expect to find a given level of
confidence.

For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find an upper bound of 0.56 and a lower bound of 0.48. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.

How do I decide which level of measurement to use?

Some variables have fixed levels. For example, gender and ethnicity are always
nominal level data because they cannot be ranked.

However, for other variables, you can choose the level of measurement. For example, income is a variable that can be recorded on an ordinal or a ratio scale:

    At an
    ordinal level, you could create 5 income groupings and code the incomes that fall within them from 1–5.At a ratio level, you would record exact numbers for income.

If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The
higher the level of measurement, the more precise your data is.

Which alpha value should I use?

The alpha value, or the threshold for statistical significance, is arbitrary – which value you use depends on your field of study.

In most cases, researchers use an alpha of 0.05, which means that there is a less than 5% chance that the data being tested could have occurred under the null hypothesis.

How do you calculate a p.-value?

P-values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p.-value tables for the relevant test statistic.

P-values are calculated from the null distribution of the test statistic. They
tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution.

If the test statistic is far from the mean of the null distribution, then the p.-value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.

What factors affect the test statistic?

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability,
the first data set will produce a test statistic closer to the null hypothesis, even if the true correlation between two variables is the same in either data set.

What is meant by model selection?

In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit
for the observed data.

The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision.

AIC model selection can help researchers find a model that explains the observed variation in their data
while avoiding overfitting.

How is AIC calculated?

The
Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters (K) used to reach that likelihood. The AIC function is 2K – 2(log-likelihood).

Lower AIC values indicate a better-fit model, and a model with a delta-AIC (the difference between the two AIC values being compared) of more than -2 is
considered significantly better than the model it is being compared to.

What is a factorial ANOVA?

A factorial ANOVA is any ANOVA that uses more than one categorical independent variable. A two-way ANOVA is a type of factorial ANOVA.

Some examples of factorial ANOVAs include:

    Testing the combined
    effects of vaccination (vaccinated or not vaccinated) and health status (healthy or pre-existing condition) on the rate of flu infection in a population.Testing the effects of marital status (married, single, divorced, widowed), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded,
    somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.

How is statistical significance calculated in an ANOVA?

In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a
statistically significant result.

Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).

If the
F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.

What is the difference
between a one-way and a two-way ANOVA?

The only difference between one-way and two-way ANOVA is the number of independent variables. A one-way ANOVA has one independent variable, while a two-way ANOVA has two.

    One-way ANOVA: Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.Two-way ANOVA: Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior,
    master’s), and race finishing times in a marathon.

All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.

How is the error calculated in a linear regression model?

Linear regression most often uses mean-square error (MSE) to calculate the error of the model. MSE is calculated by:

measuring the distance of the observed y-values from the predicted y-values each value of x;squaring each of these distances;calculating the
mean of each of the squared distances.

Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.

What is simple linear
regression?

Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both
variables should be quantitative.

For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.

What is a regression model?

A regression model is a statistical model that estimates the relationship between one
dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).

A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.

What is the difference between a one-sample t-test and a paired t-test?

A
one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or two different points in time (for example,
measuring student performance on a test before and after being taught the material).

What does a t-test measure?

A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group
means being compared, and estimates the likelihood that this difference exists purely by chance (p.-value).

Which t-test should I use?

Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a
one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test.

If you want to know only whether a difference exists, use a two-tailed test. If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test.

What is statistical significance?

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical
test. Significance is usually denoted by a p.-value, or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p. < 0.05, which means that the data is likely to occur less than 5% of the time under the
null hypothesis.

When the p.-value falls below the chosen alpha value, then we say the result of the test is statistically significant.

Which measure of central tendency is most affected by extreme very high or very low scores?

Answer and Explanation: Option A is the solution since the mean involves every point in the data set in its calculation, it becomes the measure of central tendency most susceptible to outliers or extreme values.

Which measure of central tendency is most affected by an outlier?

Mean is the only measure of central tendency that is always affected by an outlier. Mean, the average, is the most popular measure of central tendency.

Which measure of central tendency is most affected?

Of the three measures of tendency, the mean is most heavily influenced by any outliers or skewness. In a symmetrical distribution, the mean, median, and mode are all equal. In these cases, the mean is often the preferred measure of central tendency.
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