# An Intuitive Understanding of Covariance

I'm trying to intuitively understand what it means for two random variables to have non-zero covariance.

At the moment, I imagine that the two random variables (which have non-zero covariance) both depend (at least in part) on a common random variable, and this is why they have non-zero covariance.

E.g., if $U$, $V$, $Z$ and $W$ are independent random variables, and $X_1 = \frac{U + V}{Z}$ and $X_2 = W + Z$, then since both $X_1$ and $X_2$ depend on $Z$, the two variables will have non-zero covariance.

Is this a good way of intuitively thinking about covariance? If not, is there a better way which might help?

## 1 Answer

Here is my intuition on covariance:

• The covariance of $X$ and $Y$ is positive, if the two have a monotonous correlation. This means that high values of $X$ imply high values of $Y$ and vice versa.
• It is negative, if they have a negative monotonous correlation. This means that high values of $X$ imply low values of $Y$ and vice versa.
• If the covariance is zero, then there exists no monotonous correlation between $X$ and $Y$. Other correlations are still possible though.

The covariance provides the direction of a correlation between two random variables, but it does not provide the strength of it. To make the correlation comparable it has to be normalized (compare for example Pearson correlation coefficient).

• Thanks for your reply. I should have been clearer in my question - I understand what covariance is, what I don't understand is what gives random variables a non-zero covariance in the first place. – Jack Nov 3 '17 at 12:23
• Your intuition is correct. It is correct that the covariance is only nonzero if the two random variables are not independent, therefore they depend on a common random variable. But one could intuitively think that therefore if the covariance is zero, then the random variables are independent. That is not the case, that's why I provided this answer. Sorry if my point was not clear. – RoyPJ Nov 3 '17 at 12:34
• Ah I see. Thanks RoyPJ - this is helpful. – Jack Nov 3 '17 at 12:36
{"ft_lang_label":"__label__en","ft_lang_prob":0.9307275,"math_prob":0.99477214,"size":616,"snap":"2019-35-2019-39","text_gpt3_token_len":161,"char_repetition_ratio":0.15849674,"word_repetition_ratio":0.0,"special_character_ratio":0.2711039,"punctuation_ratio":0.12598425,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99975497,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-20T19:10:26Z\",\"WARC-Record-ID\":\"<urn:uuid:2b25a1b6-a21d-4d98-9f14-5f6a90cd8586>\",\"Content-Length\":\"135419\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a05b7cb4-97a9-403b-b3dd-bbbdb87f148a>\",\"WARC-Concurrent-To\":\"<urn:uuid:66c2a097-dd16-4840-be2e-9eb66c41c593>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/2502800/an-intuitive-understanding-of-covariance\",\"WARC-Payload-Digest\":\"sha1:EFSHH3YJJMUSPZT2F4E263VOUS6OKELV\",\"WARC-Block-Digest\":\"sha1:KOXHMUELAY4GYXTE33MMXKM3MU3LZBZO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027315558.25_warc_CC-MAIN-20190820180442-20190820202442-00057.warc.gz\"}"}