In the analysis of data, a correlogram is a chart of correlation statistics. For example, in time series analysis, a plot of the sample autocorrelations versus (the time lags) is an autocorrelogram. If cross-correlation is plotted, the result is called a cross-correlogram. The correlogram is a commonly used tool for checking randomness in a data set. If random, autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero. In addition, correlograms are used in the model identification stage for Box–Jenkins autoregressive moving average time series models. Autocorrelations should be near-zero for randomness; if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The correlogram is an excellent way of checking for such randomness. In multivariate analysis, correlation matrices shown as color-mapped images may also be called "correlograms" or "corrgrams". Applications The correlogram can help provide answers to the following questions: Are the data random? Is an observation related to an adjacent observation? Is an observation related to an observation twice-removed? (etc.) Is the observed time series white noise? Is the observed time series sinusoidal? Is the observed time series autoregressive? What is an appropriate model for the observed time series? Is the model valid and sufficient? Is the formula valid? Importance Randomness (along with fixed model, fixed variation, and fixed distribution) is one of the four assumptions that typically underlie all measurement processes. The randomness assumption is critically important for the following three reasons: Most standard statistical tests depend on randomness. The validity of the test conclusions is directly linked to the validity of the randomness assumption. Many commonly used statistical formulae depend on the randomness assumption, the most common formula being the formula for determining the standard error of the sample mean: where s is the standard deviation of the data. Although heavily used, the results from using this formula are of no value unless the randomness assumption holds. For univariate data, the default model is If the data are not random, this model is incorrect and invalid, and the estimates for the parameters (such as the constant) become nonsensical and invalid. Estimation of autocorrelations The autocorrelation coefficient at lag h is given by where ch is the autocovariance function and c0 is the variance function The resulting value of rh will range between −1 and