An apparatus and method for driving a liquid crystal display minimizes a gray loss of an image displayed on an RGBW-type display device, and enhances brightness and image quality. The apparatus for driving the LCD device includes: a liquid crystal panel including a plurality of unit pixels composed of 4-color sub-pixels; a data driver to transmit a video data signal to individual sub-pixels; a gate driver to transmit a scan pulse to the sub-pixels; a data converter to generate a histogram using a gray difference of input 3-color source data, to convert the 3-color source data into 4-color data according to a gain value extracted from the histogram, and to output the 4-color data; and a timing controller to transmit the 4-color data received from the data converter to the data driver and to control the gate driver and the data driver.