I have used the StackExchange API to pull some data from early this calendar year, counting questions by date.
Here is the trend:

The bottom line is that once the holiday season passed, our questions per day have gone down, and are not a huge number in general.
Our average questions per day for the analyzed period is 8.2 questions, although that is influenced by the higher peak at the beginning of the year.
Here is the detail.
1/1/2013 12
1/2/2013 10
1/3/2013 19
1/4/2013 18
1/5/2013 13
1/6/2013 17
1/7/2013 25
1/8/2013 24
1/9/2013 10
1/10/2013 15
1/11/2013 9
1/12/2013 8
1/13/2013 12
1/14/2013 21
1/15/2013 11
1/16/2013 6
1/17/2013 7
1/18/2013 7
1/20/2013 5
1/21/2013 6
1/22/2013 2
1/19/2013 3
1/24/2013 7
1/23/2013 2
1/25/2013 6
1/26/2013 7
1/27/2013 9
1/28/2013 9
1/29/2013 3
1/30/2013 7
1/31/2013 5
2/1/2013 11
2/2/2013 9
2/3/2013 2
2/4/2013 12
2/5/2013 3
2/6/2013 4
2/7/2013 4
2/8/2013 5
2/9/2013 5
2/10/2013 5
2/11/2013 7
2/12/2013 9
2/14/2013 9
2/15/2013 8
2/16/2013 5
2/13/2013 2
2/17/2013 3
2/18/2013 6
2/19/2013 6
2/20/2013 8
2/21/2013 9
2/22/2013 4
2/23/2013 9
2/24/2013 6
2/25/2013 12
2/26/2013 4
2/27/2013 7
2/28/2013 7
3/1/2013 7
3/2/2013 9
3/3/2013 7
3/4/2013 6
3/5/2013 6
3/6/2013 5
TOTAL 536
I apologize for the ugly formatting, but Markdown is decidedly data unfriendly, doesn't support table HTML tags, and I got tired of fighting with it.
I am adding the actual code used to create this data set, in case anyone sees a problem with it. Note that I ran it in LINQPAD (using the C# program option) for convenience so it doesn't look like a typical C# program. It uses the ServiceStack.Text library for JSON processing. The rest of the references are pure .NET.
void Main()
{
var baseUrl = "http://api.stackexchange.com/2.1/questions?fromdate=1356998400&todate=1362614400&order=asc&sort=activity&site=cooking&filter=!6XZr46PUugHGn&pagesize=100";
IEnumerable<Question> allQ = new List<Question>();
var more = true;
WebClient client = new WebClient();
var page = 0;
while (more) {
var content = Util.GetUrl(baseUrl + (page == 0 ? "" : "&page=" + page ));
var data = content.FromJson<Response>();
allQ = allQ.Concat(data.items);
more = data.has_more;
if (data.has_more) {
page += 1;
}
Console.WriteLine("Got page {0}, Count={1}, More={2}", page, data.items.Count(), more);
}
var analysis =
from q in allQ
group q by q.Created.HasValue ? q.Created.Value.Date : DateTime.MinValue into day
select new {
date = day.Key.ToString("d"),
count = day.Count(),
titles = from m in day select m.title
};
LINQPad.Extensions.Dump(analysis);
}
public class Response {
public Question[] items { set; get; }
public int quota_remaining { set; get; }
public int quota_max { set; get; }
public bool has_more { set; get; }
}
public class Question {
public string question_id { set; get; }
public string creation_date { set; get; }
public string last_activity_date { set; get; }
public int score { set; get; }
public int answer_count{ set; get; }
public string owner{ set; get; }
public string title{ set; get; }
public string[] tags{ set; get; }
public int view_count{ set; get; }
public string link{ set; get; }
public bool is_answered{ set; get; }
public DateTime? Created { get { return AsDate(creation_date); } }
public static DateTime? AsDate(string arg) {
long d;
var ok = long.TryParse(arg, out d);
if (!ok) { return null; }
var epoch = new DateTime(1970, 1, 1, 0, 0, 0, DateTimeKind.Utc);
return epoch.AddSeconds(d);
}
}
public class Util {
public static string GetUrl(string url) {
// Download url.
using (WebClient client = new WebClient())
{
client.Headers[HttpRequestHeader.AcceptEncoding] = "gzip";
byte[] data = client.DownloadData(url);
byte[] decompress = Decompress(data);
string text = System.Text.Encoding.UTF8.GetString(decompress);
Console.WriteLine(" URL: {0}", url);
Console.WriteLine(" Size from network: {0}", data.Length);
Console.WriteLine(" Size decompressed: {0}", decompress.Length);
Console.WriteLine(" First chars: {0}", text.Substring(0, 5));
return text;
}
}
static byte[] Decompress(byte[] gzip)
{
// Create a GZIP stream with decompression mode.
// ... Then create a buffer and write into while reading from the GZIP stream.
using (GZipStream stream = new GZipStream(new MemoryStream(gzip), CompressionMode.Decompress))
{
const int size = 4096;
byte[] buffer = new byte[size];
using (MemoryStream memory = new MemoryStream())
{
int count = 0;
do
{
count = stream.Read(buffer, 0, size);
if (count > 0)
{
memory.Write(buffer, 0, count);
}
}
while (count > 0);
return memory.ToArray();
}
}
}
}