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In statistics, the mode is the value that appears most often in a set of data values.[1] If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e, x=argmaxxi P(X = xi)). In other words, it is the value that is most likely to be sampled.

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Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population. The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

The mode is not necessarily unique to a given discrete distribution since the probability mass function may take the same maximum value at several points x1, x2, etc. The most extreme case occurs in uniform distributions, where all values occur equally frequently.

A mode of a continuous probability distribution is often considered to be any value x at which its probability density function has a locally maximum value.[2] When the probability density function of a continuous distribution has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution, so any peak is a mode. Such a continuous distribution is called multimodal (as opposed to unimodal).

In symmetric unimodal distributions, such as the normal distribution, the mean (if defined), median and mode all coincide. For samples, if it is known that they are drawn from a symmetric unimodal distribution, the sample mean can be used as an estimate of the population mode.

The mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample [1, 3, 6, 6, 6, 6, 7, 7, 12, 12, 17] is 6. Given the list of data [1, 1, 2, 4, 4] its mode is not unique. A dataset, in such a case, is said to be bimodal, while a set with more than two modes may be described as multimodal.

For a sample from a continuous distribution, such as [0.935..., 1.211..., 2.430..., 3.668..., 3.874...], the concept is unusable in its raw form, since no two values will be exactly the same, so each value will occur precisely once. In order to estimate the mode of the underlying distribution, the usual practice is to discretize the data by assigning frequency values to intervals of equal distance, as for making a histogram, effectively replacing the values by the midpoints of theintervals they are assigned to. The mode is then the value where the histogram reaches its peak. For small or middle-sized samples the outcome of this procedure is sensitive to the choice of interval width if chosen too narrow or too wide; typically one should have a sizable fraction of the data concentrated in a relatively small number of intervals (5 to 10), while the fraction of the data falling outside these intervals is also sizable. An alternate approach is kernel density estimation, which essentially blurs point samples to produce a continuous estimate of the probability density function which can provide an estimate of the mode.

Unlike mean and median, the concept of mode also makes sense for "nominal data" (i.e., not consisting of numerical values in the case of mean, or even of ordered values in the case of median). For example, taking a sample of Korean family names, one might find that "Kim" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system where a plurality determines victory, a single modal value determines the victor, while a multi-modal outcome would require some tie-breaking procedure to take place.

Unlike median, the concept of mode makes sense for any random variable assuming values from a vector space, including the real numbers (a one-dimensional vector space) and the integers (which can be considered embedded in the reals). For example, a distribution of points in the plane will typically have a mean and a mode, but the concept of median does not apply. The median makes sense when there is a linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the geometric median and the centerpoint.

For some probability distributions, the expected value may be infinite or undefined, but if defined, it is unique. The mean of a (finite) sample is always defined. The median is the value such that the fractions not exceeding it and not falling below it are each at least 1/2. It is not necessarily unique, but never infinite or totally undefined. For a data sample it is the "halfway" value when the list of values is ordered in increasing value, where usually for a list of even length the numerical average is taken of the two values closest to "halfway". Finally, as said before, the mode is not necessarily unique. Certain pathological distributions (for example, the Cantor distribution) have no defined mode at all.[citation needed][4] For a finite data sample, the mode is one (or more) of the values in the sample.

Pearson uses the term mode interchangeably with maximum-ordinate. In a footnote he says, "I have found it convenient to use the term mode for the abscissa corresponding to the ordinate of maximum frequency."

The physical nature of Super 8 film means that the stock's rough and grainy quality becomes part of the finished film's inherent aesthetic, and Corbijn's mastery of Super 8 is a key element in Strange/Strange Too and the development of Depeche Mode's visual components. The film restoration underwent a rigorous process over the course of several years with the participation of personnel involved in making the original films including Anton Corbijn. Because this Blu-Ray/DVD release was created from the original Super 8 film stock, the final result may seemingly lack the visual clarity that modern viewers associate with contemporary HD 4k reproduction. In some occasions where the original footage had deteriorated too much, the next best source was used.

Windows 11 in S mode is a version of Windows 11 that's streamlined for security and performance, while providing a familiar Windows experience. To increase security, it allows only apps from Microsoft Store, and requires Microsoft Edge for safe browsing.

Windows 11 in S mode is only available in the Windows 11 Home edition. If you have the Pro, Enterprise, or Education editions of Windows 10 in S mode, you will need to permanently switch out of S mode to upgrade to Windows 11.

To increase security and performance, Windows 11 in S mode only runs apps from the Microsoft Store. If you want to install an app that isn't available in the Microsoft Store, or if you have Windows 10 Pro, Enterprise, or Education and want to upgrade to Windows 11, you'll need to permanently switch out of S mode. There's no charge to switch out of S mode, but you won't be able to turn it back on. If you're blocked from switching and your device belongs to an organization, check with your administrator. Your organization can choose to keep all devices in S mode.

On the page that appears in Microsoft Store (Switch out of S mode or a similar page), select the Get button. After you confirm this action, you'll be able to install apps from outside Microsoft Store.

Apps you bought from Microsoft Store are linked to your Microsoft account. In most cases, if you install an app from Microsoft Store on another Windows 11 PC, you should be able to install it on your PC with Windows 11 in S mode if you sign in with the same Microsoft account.

Unless you switch out of S mode, you'll only be able to use apps from Microsoft Store on your device. There are important limitations regarding apps and switching out of S mode on Windows 11 Snapdragon devices.

Domain join is not available in Windows 11 Home in S mode. To enable Azure AD Domain join or On-premise Domain join, switch the device out of S mode and upgrade to Windows 11 Pro or Windows 11 Enterprise.

No. When you're using Windows 11 in S mode, Microsoft Edge is always the default web browser, and Bing is the default search engine. But you can access other browsers and search engines while in S mode. You can download any browser available in Microsoft Store in Windows, and you can navigate to any other search engine website.

If you want to download a browser (or any other application) that's not in Microsoft Store, you'll need to permanently switch out of S mode. After you switch, you can then set your default browser and search engine to the one you choose, but you won't be able to return to S mode again.

Yes, we recommend all Windows devices use antivirus software. Currently, the only antivirus software known to be compatible with Windows 11 in S mode is the version that comes with it: Windows Defender Security Center. Windows Defender Security Center delivers a robust suite of security features that help keep you safe for the supported lifetime of your Windows 11 device. For more info, see Windows security.

Many hardware accessories and peripherals (such as printers) that work with Windows 11 Home also work with Windows 11 in S mode, though sometimes they only offer limited functionality. For additional info, see Windows 10 in S mode accessory compatibility or contact your accessory manufacturer. 006ab0faaa