blocks|key|2076763|text|NumPy函数np.std接受一个可选参数ddof：“增量自由度”。默认情况下，这是0。将其设置为1以获得MATLAB结果：|type|unstyled|depth|inlineStyleRanges|offset|length|style|CODE|entityRanges|data|2076764|>>>+np.std([1,3,4,6],+ddof=1)
2.0816659994661326|code-block|syntax|javascript|2076765|在计算方差(其标准差是平方根)时，我们通常除以我们拥有的值的数量。|2076766|但是，如果我们从较大的分布中选择N元素的随机样本并计算方差，除以N可能会导致对实际方差的低估。要解决这个问题，我们可以将除以(the+degrees+of+freedom的数字减少到一个小于N的数字(通常是N-1)。ddof参数允许我们按照指定的量更改除数。|2076767|除非另有说明，否则NumPy将计算方差的有偏估计(ddof=0除以N)。如果您使用的是整个分布(而不是从较大分布中随机选取的值的子集)，这就是您想要的。如果给定了ddof参数，NumPy将改为除以N+-+ddof。|2076768|MATLAB的std的默认行为是通过除以N-1来校正样本方差的偏差。这消除了标准偏差中的一些(但可能不是全部)偏差。如果您在一个较大分布的随机样本上使用该函数，这很可能是您想要的。|2076769|@hbaderts的精彩回答提供了更多的数学细节。|2076770|entityMap|0|LINK|mutability|MUTABLE|url|http://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html|1|http://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics%2529)^0|7|6|L|4|16|1|1D|1|7|6|0|0|0|0|G|1|W|1|2N|1|2V|3|30|4|1R|M|1|0|P|6|X|1|29|4|2Q|8|0|7|3|K|3|0|0^^$0|@$1|2|3|4|5|6|7|14|8|@$9|15|A|16|B|C]|$9|17|A|18|B|C]|$9|19|A|1A|B|C]|$9|1B|A|1C|B|C]]|D|@$9|1D|A|1E|1|1F]]|E|$]]|$1|F|3|G|5|H|7|1G|8|@]|D|@]|E|$I|J]]|$1|K|3|L|5|6|7|1H|8|@]|D|@]|E|$]]|$1|M|3|N|5|6|7|1I|8|@$9|1J|A|1K|B|C]|$9|1L|A|1M|B|C]|$9|1N|A|1O|B|C]|$9|1P|A|1Q|B|C]|$9|1R|A|1S|B|C]]|D|@$9|1T|A|1U|1|1V]]|E|$]]|$1|O|3|P|5|6|7|1W|8|@$9|1X|A|1Y|B|C]|$9|1Z|A|20|B|C]|$9|21|A|22|B|C]|$9|23|A|24|B|C]]|D|@]|E|$]]|$1|Q|3|R|5|6|7|25|8|@$9|26|A|27|B|C]|$9|28|A|29|B|C]]|D|@]|E|$]]|$1|S|3|T|5|6|7|2A|8|@]|D|@]|E|$]]|$1|U|3|-4|5|6|7|2B|8|@]|D|@]|E|$]]]|V|$W|$5|X|Y|Z|E|$10|11]]|12|$5|X|Y|Z|E|$10|13]]]]

The NumPy function <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html"><code>np.std</code></a> takes an optional parameter <code>ddof</code>: "Delta Degrees of Freedom". By default, this is <code>0</code>. Set it to <code>1</code> to get the MATLAB result:

<pre><code>&gt;&gt;&gt; np.std([1,3,4,6], ddof=1)
2.0816659994661326
</code></pre>

To add a little more context, in the calculation of the variance (of which the standard deviation is the square root) we typically divide by the number of values we have. 

But if we select a random sample of <code>N</code> elements from a larger distribution and calculate the variance, division by <code>N</code> can lead to an underestimate of the actual variance. To fix this, we can lower the number we divide by (<a href="http://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics)">the degrees of freedom</a>) to a number less than <code>N</code> (usually <code>N-1</code>). The <code>ddof</code> parameter allows us change the divisor by the amount we specify.

Unless told otherwise, NumPy will calculate the biased estimator for the variance (<code>ddof=0</code>, dividing by <code>N</code>). This is what you want if you are working with the entire distribution (and not a subset of values which have been randomly picked from a larger distribution). If the <code>ddof</code> parameter is given, NumPy divides by <code>N - ddof</code> instead.

The default behaviour of MATLAB's <code>std</code> is to correct the bias for sample variance by dividing by <code>N-1</code>. This gets rid of some of (but probably not all of) of the bias in the standard deviation. This is likely to be what you want if you're using the function on a random sample of a larger distribution.

The nice answer by @hbaderts gives further mathematical details.

blocks|key|1863507|text|标准差是方差的平方根。随机变量X的方差定义为|type|unstyled|depth|inlineStyleRanges|offset|length|style|CODE|entityRanges|data|1863508|​|1863509|📷|atomic|1863510|1863511|因此，方差的估计值为|1863512|1863513|1863514|1863515|哪里|1863516|1863517|表示样本均值。对于随机选择的|1863518|1863519|，可以证明这个估计器不收敛于实际方差，而是收敛到|1863520|1863521|1863522|1863523|如果随机选择样本并估计样本均值和方差，则必须使用校正(无偏)估计器|1863524|1863525|1863526|1863527|它将会聚为|1863528|1863529|。修正项|1863530|1863531|也被称为贝塞尔校正。|1863532|现在，默认情况下，MATLAB+std使用校正项n-1计算无偏估计器。然而，NumPy+(正如@ajcr解释的那样)默认情况下计算无校正项的有偏估计器。参数ddof允许设置任何校正项n-ddof。通过将其设置为1，您将获得与MATLAB中相同的结果。|1863533|类似地，MATLAB允许添加第二个参数w，该参数指定“加权方案”。默认值w=0会产生校正项n-1+(无偏估计器)，而对于w=1，只有n用作校正项(有偏估计器)。|1863534|entityMap|0|IMAGE|mutability|IMMUTABLE|imageUrl|https://ask.qcloudimg.com/http-save/yehe-900000/8c8cea35d1be1b19cb66114f0a92e344.png|imageAlt|1|https://ask.qcloudimg.com/http-save/yehe-900000/d332ab51694f47b2c07255301a7d8487.png|2|https://ask.qcloudimg.com/http-save/yehe-900000/6ff93a26cc98416af767859b6df02d05.png|3|https://ask.qcloudimg.com/http-save/yehe-900000/341afa0b1e3564126cdcc96cb2f2481b.png|4|https://ask.qcloudimg.com/http-save/yehe-900000/c594b62cb038ba5f4fb3368328e254f1.png|5|https://ask.qcloudimg.com/http-save/yehe-900000/6cacb5fb86f407746b576d14b8874b19.png|6|https://ask.qcloudimg.com/http-save/yehe-900000/15f40b8a06ab66f46f185240985a4ef7.png|7|https://ask.qcloudimg.com/http-save/yehe-900000/03cfebf76949268f6d2ed39b042164be.png|8|LINK|MUTABLE|url|http://mathworks.com/help/matlab/ref/std.html^0|F|1|0|0|0|1|0|0|0|0|0|0|1|1|0|0|0|0|1|2|0|0|0|1|3|0|0|0|0|1|4|0|0|0|0|0|1|5|0|0|0|0|1|6|0|0|0|1|7|0|0|G|3|O|3|26|4|2J|6|G|3|8|0|J|1|10|3|19|3|1O|3|0^^$0|@$1|2|3|4|5|6|7|2A|8|@$9|2B|A|2C|B|C]]|D|@]|E|$]]|$1|F|3|G|5|6|7|2D|8|@]|D|@]|E|$]]|$1|H|3|I|5|J|7|2E|8|@]|D|@$9|2F|A|2G|1|2H]]|E|$]]|$1|K|3|G|5|6|7|2I|8|@]|D|@]|E|$]]|$1|L|3|M|5|6|7|2J|8|@]|D|@]|E|$]]|$1|N|3|G|5|6|7|2K|8|@]|D|@]|E|$]]|$1|O|3|I|5|J|7|2L|8|@]|D|@$9|2M|A|2N|1|2O]]|E|$]]|$1|P|3|G|5|6|7|2P|8|@]|D|@]|E|$]]|$1|Q|3|R|5|6|7|2Q|8|@]|D|@]|E|$]]|$1|S|3|I|5|J|7|2R|8|@]|D|@$9|2S|A|2T|1|2U]]|E|$]]|$1|T|3|U|5|6|7|2V|8|@]|D|@]|E|$]]|$1|V|3|I|5|J|7|2W|8|@]|D|@$9|2X|A|2Y|1|2Z]]|E|$]]|$1|W|3|X|5|6|7|30|8|@]|D|@]|E|$]]|$1|Y|3|G|5|6|7|31|8|@]|D|@]|E|$]]|$1|Z|3|I|5|J|7|32|8|@]|D|@$9|33|A|34|1|35]]|E|$]]|$1|10|3|G|5|6|7|36|8|@]|D|@]|E|$]]|$1|11|3|12|5|6|7|37|8|@]|D|@]|E|$]]|$1|13|3|G|5|6|7|38|8|@]|D|@]|E|$]]|$1|14|3|I|5|J|7|39|8|@]|D|@$9|3A|A|3B|1|3C]]|E|$]]|$1|15|3|G|5|6|7|3D|8|@]|D|@]|E|$]]|$1|16|3|17|5|6|7|3E|8|@]|D|@]|E|$]]|$1|18|3|I|5|J|7|3F|8|@]|D|@$9|3G|A|3H|1|3I]]|E|$]]|$1|19|3|1A|5|6|7|3J|8|@]|D|@]|E|$]]|$1|1B|3|I|5|J|7|3K|8|@]|D|@$9|3L|A|3M|1|3N]]|E|$]]|$1|1C|3|1D|5|6|7|3O|8|@]|D|@]|E|$]]|$1|1E|3|1F|5|6|7|3P|8|@$9|3Q|A|3R|B|C]|$9|3S|A|3T|B|C]|$9|3U|A|3V|B|C]|$9|3W|A|3X|B|C]]|D|@$9|3Y|A|3Z|1|40]]|E|$]]|$1|1G|3|1H|5|6|7|41|8|@$9|42|A|43|B|C]|$9|44|A|45|B|C]|$9|46|A|47|B|C]|$9|48|A|49|B|C]]|D|@]|E|$]]|$1|1I|3|-4|5|6|7|4A|8|@]|D|@]|E|$]]]|1J|$1K|$5|1L|1M|1N|E|$1O|1P|1Q|-4]]|1R|$5|1L|1M|1N|E|$1O|1S|1Q|-4]]|1T|$5|1L|1M|1N|E|$1O|1U|1Q|-4]]|1V|$5|1L|1M|1N|E|$1O|1W|1Q|-4]]|1X|$5|1L|1M|1N|E|$1O|1Y|1Q|-4]]|1Z|$5|1L|1M|1N|E|$1O|20|1Q|-4]]|21|$5|1L|1M|1N|E|$1O|22|1Q|-4]]|23|$5|1L|1M|1N|E|$1O|24|1Q|-4]]|25|$5|26|1M|27|E|$28|29]]]]

The standard deviation is the square root of the variance. The variance of a random variable <code>X</code> is defined as

<img src="https://i.stack.imgur.com/Gie1h.png" alt="definition of variance">

An estimator for the variance would therefore be

<img src="https://i.stack.imgur.com/aCNlB.png" alt="biased estimator">

where <img src="https://i.stack.imgur.com/yZsYN.png" alt="sample mean"> denotes the sample mean. For randomly selected <img src="https://i.stack.imgur.com/wVCga.png" alt="xi">, it can be shown that this estimator does not converge to the real variance, but to

<img src="https://i.stack.imgur.com/5JJDf.png" alt="unbiased estimator">

If you randomly select samples and estimate the sample mean and variance, you will have to use a corrected (unbiased) estimator

<img src="https://i.stack.imgur.com/NFF1W.png" alt="unbiased estimator">

which will converge to <img src="https://i.stack.imgur.com/gMDPS.png" alt="sigma squared">. The correction term <img src="https://i.stack.imgur.com/nNaCO.png" alt="n-1"> is also called Bessel's correction.

Now by default, MATLABs <a href="http://mathworks.com/help/matlab/ref/std.html" rel="noreferrer"><code>std</code></a> calculates the unbiased estimator with the correction term <code>n-1</code>. NumPy however (as @ajcr explained) calculates the biased estimator with no correction term by default. The parameter <code>ddof</code> allows to set any correction term <code>n-ddof</code>. By setting it to 1 you get the same result as in MATLAB. 

Similarly, MATLAB allows to add a second parameter <code>w</code>, which specifies the "weighing scheme". The default, <code>w=0</code>, results in the correction term <code>n-1</code> (unbiased estimator), while for <code>w=1</code>, only n is used as correction term (biased estimator).

I try to convert matlab code to numpy and figured out that numpy has a different result with the std function.

in matlab

<pre><code>std([1,3,4,6])
ans = 2.0817
</code></pre>

in numpy

<pre><code>np.std([1,3,4,6])
1.8027756377319946
</code></pre>

Is this normal? And how should I handle this?

Why does numpy std() give a different result to matlab std()?

我尝试将matlab代码转换为numpy，并计算出numpy与std函数的结果不同。在matlab中std([1,3,4,6])ans =  2.0817在numpy中np.std([1,3,4,6])1.8027756377319946这是正常的吗？我该如何处理这个问题呢？

问为什么numpy std()给出的结果与matlab std()不同？
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问为什么numpy std()给出的结果与matlab std()不同？
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