Taken verbatim from here in Google Sites help. I have italicized the most important information for automatically updating the chart.
To insert a chart into a site, follow these steps:
- Open the site page you'd like to insert a chart into.
- Click the edit page button.
- Go to the Insert menu, and select Chart.
- Select a spreadsheet from the list that contains the chart data you'd like to display.
- If the spreadsheet already contains a chart, you can select the chart. This will be a snapshot of the spreadsheet chart, and the chart won't update if spreadsheet data is modified. To insert a chart that dynamically updates, open the charts editor and select the Live option.
- If the spreadsheet doesn't already have a chart in it, enter a range of data. If you mark Live, the chart will update as spreadsheet data is modified. If you mark Snapshot, the chart won't update unless you manually do so. Learn more about creating a chart.
- Once you've selected or created a chart, click OK.
- On the next page, you can adjust the dimensions of the chart and change the title. When you've done so, click Save, and the chart will be inserted into the page.
Keep in mind:
Anyone viewing your site can view a Snapshot chart. If you've selected Live updating for your chart, only people who have access to the spreadsheet will be able to view an updated version of the original chart.