The most well known stock index is probably the Dow Jones Industrial Average. It is a price average of 30 different stocks, with each of the thirty counting equally in the computation. In that respect the Dow is different from other indices, where the weights are proportional to the capitalization of the stock, so a more heavily capitalized stock is weighted more. (And note that the capitalization does vary as the stock price varies.) For the Dow, the 30 stocks that make up the index don't change much over time, for otherwise intertemporal comparisons of index value wouldn't be very useful. In the case of the Dow, the editorial board of the Wall Street Journal would have to choose to replace one of the 30 stocks with some alternative. The stocks that are there are chosen because they are deemed representative of some sector. The S&P 500 is a broader index (ergo the 500 in the name). Note that the stocks which constitute the index do change over time, if for no other reason than because of merger and acquisition.
Indices are used as benchmarks for performance, e.g., stock so-and-so outperformed the Dow over period such-and-such by x per cent. Some variation in stock prices is due to overall stock market performance while some is due to variation within a particular industry and the remainder in variation is attributable to performance of the specific firm. Knowing how the index did gives the individual investor the ability to parse out the industry and firm-specific components to the stock return.
Dow
Changes in the S&P 500
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