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Zero-commission and discount brokers such as 'Robinhood' can be rather misleading when placing orders. Orders are often filled with a slight change in the price, relative to free real time quotes provided.

If I place a market order when my my real time quotes say the stock price is 16.20, and it comes back as being filled at 16.23, what is the reason for the difference?

Do the stock brokers interfere with the price at all, or is it simply down to the fact that the real time data was inaccurate?

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    The 'real time price' is the price the stock last traded at. – ChrisInEdmonton Oct 30 '15 at 0:27
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    ... and is in no way a guarantee that anyone will offer you the same price. You may do better, you may do worse. – keshlam Oct 30 '15 at 1:35
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The real time price you are seeing is the last traded price, when you buy or sell sell you move the market (probably by the tiniest bit) by adding volume to one side of the trade book. The price you get is the price at which the person selling to you is willing to sell to you and for stocks, depending on direction and liquidity, this will almost definitely not be the same as the last price. Remember that this is a market with people buying and selling so the price changes with each matching of a buyer to a seller.

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The market price is not a single item. It is composed of several streams of data.

market prices during normal trading

First we have the Time of Sale data, which represents each transaction that has already taken place in the market. This is the data that is used to build candle charts.

Then we have the limit order book data. This data has several parts. On the bid, we have the Buy orders that have prices lower than all the sell orders. On the Ask we have the Sell orders that have prices higher than all buy orders. The level 1 is the composed of the Best Bid and Offer in the book.

In many markets like the NYSE and NASDAQ there are many order books that compete with each other. Each of them keeping a set of bids and offers available for execution. The best prices of the combined books is known as the National Best Bid and Offer or NBBO.

order execution

When you send a market order to buy. Your order is executed against the limit sell orders that are available to sell at the time of your order's arrival. It will execute first against orders at a lower price, and if your order is too large to filled at a single price it will continue taking out sell orders at subsequently higher prices.

your particular scenario

In your particular case, your broker may be showing you as the "real time price" the price of the last sale, the price of the bid (many forex brokers do this). If they are showing you the ask price, then you would notice that you are able to consistently sell at the real time price, if they use the TOS price then you will see the price flickering up and down 2 or 3 cents as the orders come in to buy or sell.

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