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Related Concept Videos

Harmonic Mean01:09

Harmonic Mean

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The arithmetic mean is usually skewed towards the larger values in the data set. Therefore, to avoid this inherent bias towards smaller values, the harmonic mean is used.
Take the example of the speed of a car, which is the measure of the rate of distance traveled. If the vehicle traverses the same distance back-and-forth, its average speed equals the total distance traveled divided by the total time taken. However, if the car moves with varying speeds, then the arithmetic mean is more skewed...
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Trimmed Mean01:10

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While measuring the mean of a data set, care needs to be taken when associating the mean to its central tendency. The same goes for the arithmetic mean, the geometric mean, or the harmonic mean. This is because the presence of a single outlier data value can significantly affect the mean. That is, the mean is sensitive to fluctuations in the data set.
Although certain measures of central tendency are not sensitive to outliers, there are alternative versions of the mean that get around the...
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Root Mean Square00:57

Root Mean Square

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If in an experiment, data values have a probability of being both positive and negative, neither the arithmetic mean, the geometric mean, nor the harmonic mean can be used to calculate the central tendency of the data set. In particular, if the positive and negative values are equally likely, the arithmetic mean is close to zero.
For example, consider the velocity of gas molecules in a container. The gas molecules are moving in different directions, which might impart positive and negative...
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Mean Absolute Deviation01:13

Mean Absolute Deviation

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The mean absolute deviation is also a measure of the variability of data in a sample. It is the absolute value of the average difference between the data values and the mean.
Let us consider a dataset containing the number of unsold cupcakes in five shops: 10, 15, 8, 7, and 10. Initially, calculate the sample mean. Then calculate the deviation, or the difference, between each data value and the mean. Next, the absolute values of these deviations are added and divided by the sample size to...
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Weighted Mean00:57

Weighted Mean

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While taking the arithmetic, geometric, or harmonic mean of a sample data set, equal importance is assigned to all the data points. However, all the values may not always be equally important in some data sets. An intrinsic bias might make it more important to give more weightage to specific values over others.
For example, consider the number of goals scored in the matches of a tournament. While computing the average number of goals scored in the tournament, it may be more important to...
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Skewness01:06

Skewness

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The measures of central tendency calculated from a data set may not reveal much about its intrinsic distribution. If a plot is made of the data set’s values, the mean and the median may not only differ, but also the plot may have more values on one side of the central tendencies. Such a data set is said to be skewed towards that side.
The longer the tail of the plot on one side, the more skewed it is. The skewness of a data set’s values suggests that the measures of central tendency...
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Related Experiment Video

Updated: Jun 29, 2025

Magnetic Resonance Derived Myocardial Strain Assessment Using Feature Tracking
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Convergence Analysis of Mean Shift.

Ryoya Yamasaki, Toshiyuki Tanaka

    IEEE Transactions on Pattern Analysis and Machine Intelligence
    |April 8, 2024
    PubMed
    Summary

    The mean shift (MS) algorithm finds the peak of a data distribution. This study proves its convergence and analyzes its speed for kernel density estimation (KDE), enhancing mode estimation accuracy.

    Area of Science:

    • Statistics
    • Machine Learning
    • Data Analysis

    Background:

    • The mean shift (MS) algorithm is a fundamental tool for identifying modes in kernel density estimates (KDE).
    • Existing research provides some convergence properties but lacks guarantees for broader kernel types.

    Purpose of the Study:

    • To establish a convergence guarantee for the sequence of mode estimates generated by the MS algorithm.
    • To evaluate the convergence rate of the MS algorithm under mild conditions.
    • To extend the theoretical understanding of MS algorithm convergence beyond standard kernels.

    Main Methods:

    • Utilizing the Łojasiewicz inequality argument to analyze the convergence properties.
    • Applying theoretical analysis to kernel density estimation (KDE) mode-seeking algorithms.

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  • Extending convergence proofs to include analytic and Epanechnikov kernels.

    • Main Results:

    • A formal convergence guarantee for the MS algorithm's mode estimate sequence is established.
    • The convergence rate of the MS algorithm is evaluated under mild conditions.
    • The study extends existing convergence results to cover analytic and Epanechnikov kernels.

    Conclusions:

    • The findings provide a theoretical foundation for the reliability of the MS algorithm in mode estimation.
    • The inclusion of the biweight kernel, optimal for asymptotic statistical efficiency, is a significant extension.
    • This work enhances the understanding and applicability of MS algorithm for KDE-based mode estimation.