One kilobit is 0.125 kilobytes — one eighth, the same factor of eight that separates every bit unit from its byte counterpart. At this scale the pair belongs to audio encoding and to the history of dial-up.
About the units
A kilobit is a thousand bits and a kilobyte a thousand bytes, so the ratio is exactly eight. The kilobit per second was the defining unit of consumer networking for two decades: dial-up modems progressed from 300 bit/s through 14.4, 28.8 and finally 56 kbit/s, and each step was a headline. That last figure was never achievable in practice — regulatory limits on line voltage capped real downstream rates at about 53 kbit/s, which is 6.6 kB/s, or roughly a minute and a half for a single modern web page.
The exact factor
The factor is exactly 0.125. A 56 kbit/s modem moves 7 kB/s at its theoretical best. In audio, a 128 kbit/s MP3 consumes 16 kB per second of playback, so a four-minute track is about 3.8 MB — the arithmetic that made portable music players practical in the early 2000s. A 320 kbit/s file is 40 kB/s and 9.6 MB for the same track.
Where you meet this conversion
Audio and video encoding above all. Bitrate settings in an encoder are in kilobits per second, and the resulting file size in kilobytes or megabytes, so anyone choosing a quality setting is doing this conversion whether they notice it or not. Voice codecs are specified in kilobits: a mobile call uses about 13 kbit/s, and voice over IP codecs range from 8 to 64. Embedded and industrial networking still operates at kilobit rates, where CAN bus and Modbus links are specified in kbit/s.